vector bundles and complex geometry

CONTEMPORARYMATHEMATICS

American Mathematical Society

522

Vector Bundles and Complex Geometry

Conference on Vector Bundles in Honor of S. Ramanan on the Occasion

of his 70th BirthdayJune 16–20, 2008

Miraflores de la Sierra, Madrid, Spain

Oscar García-PradaPeter E. NewsteadLuis Álvarez-Cónsul

Indranil BiswasSteven B. BradlowTomás L. Gómez

Editors

This page intentionally left blank

American Mathematical SocietyProvidence, Rhode Island

CONTEMPORARYMATHEMATICS

522


Conference on Vector Bundles in Honor of S. Ramanan on the Occasion

of his 70th Birthday June 16–20, 2008

Miraflores de la Sierra, Madrid, Spain

Oscar García-Prada Peter E. Newstead Luis Álvarez-Cónsul

Indranil Biswas Steven B. Bradlow Tomás L. Gómez

Editors

Editorial Board

Dennis DeTurck, managing editor

George Andrews Abel Klein Martin J. Strauss

2000 Mathematics Subject Classification. Primary 14H60, 14D20, 20G15, 14D07, 14D22,58J20, 14C30, 14J60.

Library of Congress Cataloging-in-Publication Data

Vector bundles and complex geometry : conference on vector bundles in honor of S. Ramananon the occasion of his 70th birthday, June 16–20, 2008, Miraflores de la Sierra, Madrid, Spain /Oscar Garcıa-Prada . . . [et al], editors.

p. cm. — (Contemporary mathematics ; v. 522)Includes bibliographical references and index.ISBN 978-0-8218-4750-3 (alk. paper)1. Vector bundles—Congresses. 2. Geometry, Algebraic—Congresses. I. Ramanan, S.

II. Garcıa-Prada, O. (Oscar), 1960–

QA612.63.V418 2010514′.224—dc22

2010011114

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10 9 8 7 6 5 4 3 2 1 15 14 13 12 11 10

To S. Ramanan on the occasion of his 70th birthday

Contents

Preface ix

The Work of S. RamananM. S. Narasimhan 1

Parabolic Bundles on Algebraic Surfaces II - Irreducibility of the Moduli SpaceV. Balaji and A. Dey 7

Finite Subgroups of PGL2(K)Arnaud Beauville 23

Picard Groups of Moduli Spaces of Torsionfree Sheaves on CurvesUsha N. Bhosle 31

On the Moduli of Orthogonal Bundles on a Nodal Hyperelliptic CurveUsha N. Bhosle 43

Hilbert Schemes of Fat r-Planes and the Triviality of Chow Groups ofComplete Intersections

Andre Hirschowitz and Jaya NN Iyer 53

Vector Bundles and the IcosahedronNigel Hitchin 71

Cohomology of the Toroidal Compactification of A3

Klaus Hulek and Orsola Tommasi 89

Quasi-Complete Homogeneous Contact Manifold Associated to a Cubic FormJun-Muk Hwang and Laurent Manivel 105

Maximal Weights in Kahler Geometry: Flag Manifolds and Tits Distance (withan Appendix by A. H. W. Schmitt)

I. Mundet i Riera 113

Orthogonal Bundles Over Curves in Characteristic TwoChristian Pauly 131

The Atiyah-Singer Index TheoremM. S. Raghunathan 141

Spin(7) Instantons and the Hodge Conjecture for Certain Abelian Four-folds:A Modest Proposal

T. R. Ramadas 155

vii

viii CONTENTS

Remarks on Parabolic StructuresC. S. Seshadri 171

Iterated Destabilizing Modifications for Vector Bundles with ConnectionCarlos Simpson 183

Preface

This volume is dedicated to S. Ramanan on the occasion of his 70th birthday.Ramanan’s influence on mathematics is the common thread running through allthe articles in this volume. In some cases it is seen directly in the subject matterof the research, while in other cases it is through his association—as colleague ormentor—with the author.

Ramanan has made important contributions to Differential and Algebraic Ge-ometry. He has been a leading expert on vector bundles and moduli spaces forover 40 years. For many years a Distinguished Professor at the Tata Institute ofFundamental Research in Mumbai, he is now an honorary professor at the Instituteof Mathematical Sciences in Chennai and Adjunct Professor at the Chennai Mathe-matics Institute. He is a Fellow of all three Indian National Science Academies andis a recipient of the Shanti Swarup Bhatnagar Prize, the Srinivasa Ramanujan Birthcentenary award and the Third World Academy of Sciences Prize for Mathematics,among other things. Some of his many profound contributions are described in thearticle in this volume by M.S. Narasimhan.

Most of the articles come from a workshop on vector bundles, held in theMiraflores conference center outside Madrid, Spain in June 2008. In addition tobeing a felicitation for Ramanan, the workshop served several purposes:

• It was the 2008 annual workshop of the Vector Bundles on AlgebraicCurves (VBAC) group. Founded in 1994 by Peter Newstead, the VBACgroup has been an influential champion of vector bundles and their evolv-ing place in mathematics. Ramanan is not only an early member of therenowned ‘Tata school’ of vector bundles, but has been a central memberof VBAC from its first days.

• The workshop was part of a semester-long program on Moduli Spaces,organized by CSIC (Madrid), Universidad del Paıs Vasco, Universidad deSalamanca and Universitat de Barcelona. Other activities in the programincluded an International School on Geometry and Physics: moduli spacesin geometry, topology and physics (Cantabria, 25-29 February 2008), aworkshop on Moduli spaces of vector bundles: algebro-geometric aspects(Barcelona, 12-14 March 2008), and a workshop on Bundles, gerbes andderived categories in string theory (Salamanca, 14-16 May 2008)

Finally, no description of the Miraflores workshop would be complete withouta mention of the evening of classical Indian song, with performances by workshopparticipants Kavita Ramanan, Tomas Gomez, V. Balaji, C.S. Seshadri, and Ra-manan himself. With its minimum of fuss but abundance of beauty and skill, thememorable evening was a perfect metaphor for Ramanan’s contribution to mathe-matics.

ix

Contemporary Mathematics

The Work of S. Ramanan

M.S. Narasimhan

Abstract. This talk gives an overview of some of the significant work of S.Ramanan in the areas of moduli problems, linear systems on flag varieties andabelian varieties, and differential geometry.

1. Introduction

It is with great pleasure that I write this account of the work of S.Ramanan.I have been fortunate enough to have collaborated with him intensely over a longperiod. I have profited much from his mathematical insights.

His mathematics is characterised by depth and breadth and covers several areas:algebra, differential geometry and algebraic geometry. In algebraic geometry he isa major figure, and his main interests have been:

• moduli of vector bundles on curves,• homogeneous vector bundles on flag varieties,• abelian varieties,• geometric invariant theory,• linear systems on flag varieties in positive characteristics,• Green’s conjecture on syzygies of canonical curves, and• Higgs bundles.

His contributions to the study of vector bundles on curves have been particu-larly extensive and profound.

I will describe below some of Ramanan’s work to give an idea of the significanceand the impact of his contributions.

2. Universal connections

The basic theorem on the existence of universal connections was proved byRamanan and myself in [1] and [2]. (This was our first collaboration.)

Theorem 2.1. Let G be a Lie group with a finite number of connected compo-nents, and n a positive integer. Then there exists a principal G-bundle En and aconnection ωn on En, such that any connection on a principal G-bundle π : P → Mwith dim M ≤ n is the inverse image of ωn by a G-morphism of P into En.

2010 Mathematics Subject Classification. 14D20, 14D22, 14K25, 14L24, 53C05, 20G15.

c©0000 (copyright holder)

1

Contemporary MathematicsVolume 522, 2010

c©2010 American Mathematical Society

1

2 M.S. NARASIMHAN

When G = U(k) (the unitary group), En may be taken to be the Stiefel mani-fold V (N, k) of unitary k-frames in Cn and ωn to be the canonical connection A∗dA,with A = the N × k matrix representing a point of V (N, k).

This theorem has been used extensively by mathematicians and physicists:in the work of Chern-Simons, in the definition of the Cheeger-Simons differentialcharacter, in Quillen’s work on super-connections and more recently in the contextof stochastic differential equations.

3. Homogeneous vector bundles

The work [3] constituted Ramanan’s thesis.Let G be a semi-simple algebraic group over C. Let P be a parabolic subgroup

of G. (Let us assume that P does not contain a simple component of G). Letρ be a finite-dimensional irreducible representation of G and Vρ the associatedholomorphic vector bundle over G/P .

Theorem 3.1. (1) If ρ1 and ρ2 are irreducible representations of P then Vρ1is

isomorphic to Vρ2(as holomorphic bundles) if and only if ρ1 and ρ2 are equivalent

representations.(2) Vρ is a simple bundle; in particular, Vρ is indecomposable.(3) In fact, for any polarisation on G/P, Vρ is a μ-stable vector bundle.

The equivalence problem had been raised by Ise (1960). The statement (3) wasproved first in the case of an irreducible symmetric space. Umemura observed that(3) is valid in general.

Ramanan also showed that there are homogeneous vector bundles Vρ on P2 withH2(P2, EndVρ) = 0 and H1(P2, EndVρ) �= 0. (These dimensions can be calculated).

4. Moduli of vector bundles on curves

Ramanan has done extensive and celebrated work in this area. I shall describebelow some of his contributions in this field.

Let X be a smooth irreducible curve over C, of genus g ≥ 2; let ξ denote aline bundle on X of degree d. Let U(r, d) (respectively, S(r, ξ)) denote the modulispace of (semi-stable) vector bundles on X of rank r and degree d (respectively,with determinant isomorphic to ξ). We denote S(r, ξ) also by S(r, d). The followingwas proved in [4]:

Theorem 4.1. The set of non-singular points of U(r, d) is precisely the set ofstable points in U(r, d) except when g = 2, r = 2 and d even.

Let S(2, 0) be the space of semi-stable bundles of rank 2 and trivial determinant.In [4] and [5] we proved:

Theorem 4.2. Let g = 2.(1) S(2, 0) is isomorphic to P

3(C).(2) If ξ is a line bundle of degree 1, S(2, ξ) is isomorphic to a smooth intersec-

tion of two smooth quadrics in P5.

Let Jg−1 be the space of line bundles of degree (g−1) on X and θ the canonicaltheta divisor on Jg−1. For any semi-stable vector bundle E of rank 2 and trivialdeterminant letDE denote the subset of Jg−1 defined byDE = {ξ εJg−1, H0(X, ξ⊗E) �= 0}. Then DE is a divisor linearly equivalent to 2θ and defines a morphism fof S := S(2, 0) into the projective space |2θ|. In [4] we proved:

2

THE WORK OF S. RAMANAN 3

Theorem 4.3. For g = 2, f is an isomorphism.

Turning to the case of genus 3, we have ([6]):

Theorem 4.4. Let g = 3, and X non-hyperelliptic. Then S is isomorphicto a quartic hypersurface in P7. (In fact the morphism f defined above gives theimbedding).

This quartic had been considered earlier by Coble. In fact the above theoremhelps to solve a problem posed by Coble.

Note that non-stable points in S correspond to vector bundles of the formL⊕L−1, where L is a line bundle of degree 0. That is, the set of non-stable pointsis the Kummer variety associated to the Jacobian of X. In the case g = 2 theKummer surface is imbedded in P

3 and was originally investigated by Kummer fromthe point of view of quadratic complexes of lines in P3. (A quadratic complex Q isthe intersection of two quadrics in P5, where one of the quadrics is the grassmannianof lines in P3). The incidence correspondence between lines and points in P3 inducesa diagram:

HP1−bundle

��conic bundle

��

��

�

Q = S(2, ξ) P3

The P1 bundle over P3 −K does not come from a vector bundle. This impliesthe non-existence of a Poincare family over the open subset of stable points inS = S(2, 0), (g = 2, r = 2, d = 0).

To prove that the P1 - bundle does not come from a vector bundle one interpretsthe incidence correspondence in terms of vector bundles.

In general one gets the Hecke correspondence, between moduli spaces. Forinstance when r = 2 one gets :

HP1−bundle

��conic bundle

��

S(2, 1) S(2, 0)

The Hecke correspondence is the key to the study of the geometry of modulispaces (and has since been used extensively by several mathematicians). Using thiscorrespondence one proves, for example, that the number of moduli of S(r, ξ) is thesame as that of the curve, when (r, d) = 1. More precisely ([7]):

Theorem 4.5. (1) The group of automorphisms of S(r, ξ)((r, d) = 1) is finiteand Hi(S(r, ξ), T ) = 0 for i �= 1, where T is the tangent sheaf.

(2) dim H1(S(r, ξ), T ) = 3g − 3.

Theorem 4.6. The canonically polarized intermediary Jacobian of S(r, ξ)((r, d) = 1) corresponding to the third cohomology group is naturally isomorphicto the canonically polarized Jacobian of X.

The Hecke correspondence was also used ([8]) to construct an explicit desingu-larisation of the moduli space of rank 2 bundles with trivial determinant.

In the paper [9] Ramanan proved a number of important results, among them:

3

4 M.S. NARASIMHAN

(1) The anticanonical class of S(r, ξ) in the case (r, d) = 1 is twice the amplegenerator of the Picard group.

(2) If r and d are not coprime, there does not exist a Poincare family on anynon-empty open subset of U(r, ε), for any rank r and genus.

(3) The structure of the cohomology ring of S(2, 1), when g = 3.

The paper [10] studies the Hitchin fibration with application to the generalisedtheta divisor.

In a paper ([11]) with Desale, Ramanan proved:

Theorem 4.7. Let X be an hyperelliptic curve of genus g ≥ 2 and λ0, ......λ2g+1

be the branch points of X in P1. Then S(2, 1) is isomorphic to the space of all(g − 2) dimensional linear subspaces of P2g+1 which are in the intersection of thetwo quadrics

Σ2g+1i=0 x2

i = Σ2g+1i=0 λi x

2i = 0

Ramanan interpreted linear subspaces of other dimensions contained in thesequadrics as a kind of spin bundles ([12]).

5. Ample divisors on abelian varieties

Theorem 5.1. [13] Let A be an abelian surface not containing elliptic curvesand L an ample bundle of type (δ1, δ2), δ1|δ2. Then L is very ample if

δ1 = 1, δ2 ≥ 5; δ1 = 2, δ2 ≥ 4 or δ1 ≥ 3

In particular L of type (1,5) embeds A in P4 (as a subvariety of codimension2) and the Serre construction yields the Horrocks-Mumford bundle on P

4.

6. Geometric Invariant Theory: Semi-stability of bundles obtained byextension of structure group

Ramanan and Ramanathan made an incisive study ([14]) of the instability flagsarising in Geometric Invariant Theroy. They used this to give an algebraic proof ofthe following result:

Let Char k = 0, k = k. Let G and H be reductive groups and ρ : G → Ha homomorphism which maps the connected component of the centre of G intothat of H. If E is a semi-stable G-bundle, then the extended H-bundle E(H) issemi-stable. If E is quasi-stable, so is the extended bundle.

If Char k > 0, their method enables one to analyse the semistability of E(H).Coiai-Holla used this method to prove the boundedness of semi-stable bundles inpositive characteristic.

7. Projective normality of flag varieties with Schubert varieties

Ramanan and Ramanathan proved ([15]):

Theorem 7.1. Let k be an algebraically closed field of arbitrary characteristicand G a reductive group over k. Let Q be a parabolic subgroup of G and L anample line bundle over G/Q. Then the complete linear system of L embeds G/Qas a projective normal variety. In particular, Schubert varieties are normal.

4

THE WORK OF S. RAMANAN 5

8. Higgs Bundles

More recently Ramanan has been interested in Higgs bundles [16]. With O.Garcıa-Prada he studied Higgs bundles twisted by a line bundle (of finite order)and determined the corresponding Tannaka group in terms of the pro-reductivecompletion of the fundamental group and the character defined by the line bundle[17]. They also study involutions on the Higgs moduli space on a curve [16]

I have not gone into many other contributions of Ramanan, for example, hisexplanation of the mysterious Capelli identity in terms of an element of the universalenveloping algebra of the linear group or his beautiful book,“Global Analysis” ([18])which gives an insightful perspective on basic differential analysis and geometry.

References

[1] M.S. Narasimhan and S. Ramanan, Existence of universal connections, Amer. J. Math. 83(1961) 563–572.

[2] M.S. Narasimhan and S. Ramanan, Existence of universal connections, II, Amer. J. Math.85 (1963) 223–231.

[3] S. Ramanan, Holomorphic vector bundles on homogeneous spaces, Topology 5 (1966) 159–177.

[4] M.S. Narasimhan and S. Ramanan, Moduli of vector bundles on a compact Riemann surface,Ann. of Math. (2) 89 (1969) 14–51.

[5] M.S. Narasimhan and S. Ramanan, Vector bundles on curves, Algebraic Geometry (Internat.Colloq.), Tata Inst. Fund. Res., Bombay, (1968) 335–346 Oxford Univ. Press, London.

[6] M.S. Narasimhan and S. Ramanan, 2θ-linear systems on abelian varieties, Vector bundles onalgebraic varieties, pp 415–427, Bombay, 1984.

[7] M.S. Narasimhan and S. Ramanan, Deformations of the moduli space of vector bundles overan algebraic curve, Ann. Math. (2) 101 (1975) 391–417.

[8] M.S. Narasimhan and S. Ramanan, Geometry of Hecke cycles. I in C. P. Ramanujam – atribute, pp. 291–345, Tata Inst. Fund. Res. Studies in Math., 8, Springer, Berlin-New York,1978.

[9] S. Ramanan, The moduli spaces of vector bundles over an algebraic curve, Math. Ann. 200(1973) 69–84.

[10] A. Beauville, M.S. Narasimhan and S. Ramanan, Spectral curves and the generalised thetadivisor, J. Reine Angew. Math. 398 (1989) 169–179.

[11] U.V. Desale and S. Ramanan, Classification of vector bundles of rank 2 on hyperellipticcurves, Invent. Math. 38 (1976/77) 161–185.

[12] S. Ramanan, Orthogonal and spin bundles over hyperelliptic curves, Proc. Indian Acad. Sci.Math. Sci. 90 (1981) 151–166.

[13] S. Ramanan, Ample divisors on abelian surfaces, Proc. London Math. Soc. (3) 51 (1985)231–245.

[14] S. Ramanan and A. Ramanathan, Some remarks on the instability flag, Tohoku Math. J. (2)36 (1984) 269–291.

[15] S. Ramanan and A. Ramanathan, Projective normality of flag varieties and Schubert vari-eties, Invent. Math. 79 (1985) 217–224.

[16] S. Ramanan, Some aspects of the theory of Higgs pairs, “The The Many Facets of Geometry:A Tribute to Nigel Hitchin”, Editors: O.Garcıa-Prada, J-P. Bourguignon and S. Salamon,Oxford University Press, 2010.

[17] O. Garcıa-Prada and S. Ramanan, Twisted Higgs bundles and the fundamental group ofcompact Kahler manifolds, Mathematical Research Letters, 7 (2000), 1–18.

[18] S. Ramanan, Global calculus. Graduate Studies in Mathematics, 65. American MathematicalSociety, Providence, RI, 2005.

TIFR, Centre for Applicable Mathematics, Bangalore 560065; and Department of

Mathematics, IISc, Bangalore 560012, India

5


Parabolic Bundles on Algebraic Surfaces II - Irreducibility ofthe Moduli Space

V. Balaji and A. Dey

To Professor Ramanan on his 70th birthday

Abstract. In this paper we prove irreducibility of the moduli space of par-abolic rank 2 bundles over an algebraic surface for c2 � 0 and with an irre-ducible parabolic divisor D of X. This gives parabolic analogues of theoremsof O’Grady and Gieseker-Li.

1. Introduction

Let X be a smooth projective surface over the ground field C of complex num-bers and let D be a smooth irreducible divisor. Let H be a very ample line bundleon X which we fix throughout. We study bundles with c1 = 0 in this paper.

We denote by Mαk,d the moduli space of parabolic H–stable parabolic bundles of

rank 2 with parabolic structure on D together with rational weights α := (α1, α2)(see (2.4) and (2.5) for the definition of the invariant d) and where k stands for thesecond Chern class c2 of a vector bundle.

In [3], the Donaldson-Uhlenbeck compactification Mαk,d of the moduli space

Mαk,d was constructed as a projective variety by realizing it as the closure of Mα

k,d

in a certain projective scheme Mαk,d endowed with the reduced scheme structure; it

was also shown to be non-empty for large k. There are also obvious bounds on theinvariant d for quasi-parabolic structures to exist.

Let MH(2, c1, c2) (resp. MH(2, c1, c2)s) denote the moduli space of H-semi-

stable (resp. stable) torsion free sheaves of rank 2 whose Chern classes are c1 andc2 respectively.

Since the topological type of the bundles is fixed for the problem as also is theample polarization H, we will have the following convenient notations:

Msα := Mα

k,d; Mα := Mαk,d

andMs := MH(2, 0, c2)

s.

2010 Mathematics Subject Classification. 14D20, 14D23.Key words and phrases. vector bundle, parabolic vector bundle, moduli space, moduli stack.Research of the first author was partly supported by the J.C. Bose Research grant.


1



7

2 V. BALAJI AND A. DEY

We carry the weight tuple α as a part of the notation since this parameter willbe varied in the arguments and the moduli spaces will be compared for differingweights.

We say a moduli space as above is asymptotically irreducible if it is irreduciblefor c2 � 0, i.e the second Chern class of the bundle underlying the parabolicbundle is large. In particular we do not quantify c2 when we address the questionof asymptotic irreducibility.

In this paper we prove asymptotic irreducibility of the moduli space Mα whenobvious bounds are imposed on d for the existence of quasi-parabolic structures.These moduli spaces for rank 2 have been studied from a differential geometricstandpoint in [12] where k = c2 stands for the “instanton number”.

Our theorem generalizes the theorem of Gieseker-Li and O’Grady ([6] and [16])to the parabolic case. The parabolic case has been of independent interest; forexample, in [14] Maruyama has shown links between the parabolic moduli spacesfor special parabolic weights and the moduli space of instantons. Maruyama usesthese links to prove irreducibility of some of these spaces.

The assumptions on the parabolic divisor, rank and full flag quasi-parabolicstructure can be relaxed; one could take the parabolic divisor to be a divisor withsimple normal crossings and the bundles to be of arbitrary rank and any quasi-parabolic type. We have made the special choices to make the paper more readable.The choice of rational weights is the natural one and real weights are really anartifice and do not occur in the classical setting. In any case this is not a seriousissue as far as the question of irreducibility of the moduli space is concerned sincethe “yoga of parabolic weights” allows us to deduce geometric statements for modulispaces with real weights from those with nearby rational weights.

The assumption on large second Chern class is what makes the statement anasymptotic one; the result is shown only for large c2.

Acknowledgements: We wish to thank S. Bandhopadhyay for many helpful discussions

while this paper was getting prepared. The second author wishes to thank the Institute

of Mathematical Sciences, Chennai and the Chennai Mathematical Institute for their

hospitality while this work was being done. Finally we wish to express our grateful thanks

to the diligent referee for correcting the innumerable errors in the earlier versions and

helping us improve the exposition.

2. Preliminaries

Our basic tool is the Seshadri-Biswas correspondence between the category ofparabolic bundles on X and the category of Γ–bundles on a suitable Kawamatacover. This strategy has been employed in several papers. Most of the materialwritten in this section is taken from §2 of [3] and the reader will find details of theSeshadri-Biswas correspondence in this reference. However in this note we are onlyinterested in the rank 2 case, and we will give definitions in the rank 2 case aloneand lay stress on those points which are relevant to our purpose.

2.1. The category of bundles with parabolic structures. Let X bea smooth projective surface over the ground field C and let D be an irreduciblesmooth divisor in X. Let H be a very ample line bundle on X.

8

PARABOLIC BUNDLES 3

Definition 2.2. Let E be a rank 2 torsion-free sheaf on X. A parabolicstructure (with respect to D) on E is a filtration (quasi–parabolic structure)

(2.1) E∗ : E = F1(E) ⊃ F2(E) ⊃ F3(E) = E ⊗OX(−D)

together with a system of weights

0 ≤ α1 < α2 < 1

where αi is the weight associated with F i(E).

(See [12, Section 8] where the weights are given in [− 12 ,

12 ) following the balanced

convention.)We will use the notation E∗ to denote a parabolic sheaf and by E (without the

subscript “∗”) when it is without its parabolic luggage. The notation E∗ thereforecarries the data of the weight tuple α as well. A parabolic sheaf E∗ is called aparabolic bundle if the underlying sheaf E is a vector bundle.

2.3. Some assumptions. The class of parabolic vector bundles that are dealtwith in the present work satisfy certain constraints which will be explained now.

(2.2) All parabolic weights are rational numbers.

(2.3) F1(E)/F2(E) is torsion-free as a sheaf on D.

We need to impose these in order to have the Seshadri–Biswas correspondence (cf.[3, Remark 2.3] for details). Henceforth, all parabolic vector bundles will be assumedto have the constraints (2.2) and (2.3).

Also note that the filtration (2.1) is equivalent to a filtration on E |D given by

(2.4) E |D= F1D(E) ⊃ F2

D(E) ⊃ F3D(E) = 0.

To see this, simply define

F i(E) = ker

(E → E |D

F iD(E)

).

In the notation Mαk,d in the introduction, the numerical invariant d is given by

(2.5) d = c1(F2D(E)).D.

The slope of a rank 2 parabolic sheaf E∗ is defined as

(2.6) μα(E∗) =[c1(E) + (α1 + α2)D] ·H

2.

Let PVect(X,D) denote the category whose objects are rank 2 parabolic vectorbundles over X with parabolic structure over the divisor D satisfying (2.2) and(2.3), and whose morphisms are homomorphisms of parabolic vector bundles (see[3] for more detail).

For an integer N ≥ 2, let PVect(X,D,N) ⊆ PVect(X,D) denote the subcat-egory consisting of all parabolic vector bundles all of whose parabolic weights aremultiples of 1/N .

Let E∗ be a rank 2 parabolic bundle on X with parabolic weight (α1, α2). LetL be a line subbundle of E, the underlying bundle of the parabolic bundle E∗. Theparabolic weights on E∗ induces a parabolic weight on L denoted by αL; αL equals

9


α2 if L ⊂ F2D(E), and is α1 otherwise. Denote this parabolic line bundle with this

induced structure by L∗. The slope of this parabolic line bundle L∗ is given by

(2.7) μαL(L∗) = (c1(L) + αLD) ·H.

It is not hard to check that for the purposes of stability it suffices to worry aboutparabolic line subbundles L∗ of a rank 2 parabolic bundle E∗ which are obtainedfrom a line subbundle L of E with a weight αL defined as above. The parabolicbundle E∗ is α-stable (resp. α-semi–stable) if

(2.8) μαL(L∗) < μα(E∗) (resp. ≤)

for all parabolic line subbundles L∗ of E∗.

2.4.The Kawamata covering lemma. LetD ⊂ X be an irreducible divisor.Take any E∗ ∈ PVect(X,D) such that all the parabolic weights of E∗ are multiplesof 1/N , i.e. E∗ ∈ PVect(X,D,N). The “covering lemma” of Y. Kawamata ([11,Theorem 1.1.1], [10, Theorem 17]) says that there is a connected smooth projectivevariety Y over C and a Galois covering

(2.9) p : Y −→ X

such that the reduced divisor D := (p∗D)red is a normal crossing divisor of Y and

furthermore the pull-back p∗D equals kND, for some positive integer k. Let Γdenote the Galois group for the covering map p (2.9).

2.5. The category of Γ–bundles. Let Γ ⊆ Aut(Y ) be a finite subgroup of

the group of automorphisms of a connected smooth projective variety Y/C and Hbe a fixed polarization on Y .

A Γ–vector bundle V on Y is a vector bundle V together with a collection ofisomorphisms of vector bundles

g : V −→ (g−1)∗V

indexed by g ∈ Γ and satisfying the condition that gh = g ◦ h for any g, h ∈ Γ (see§2, [3] for more detail).

A Γ–homomorphism between two Γ–vector bundles is a homomorphism be-tween the two underlying vector bundles which commutes with the Γ–actions. LetVectΓ(Y ) denote the category of Γ–vector bundles on Y with the morphisms beingΓ–homomorphisms.

Having fixed the parabolic divisor and the Kawamata cover together with theramification indices, one has the concept of local type of a Γ–bundle which is de-scribed in [3, 2.4.1] (see [17] for the terminology). This is needed in order to setup the correspondence between Γ–bundles and parabolic bundles with specifiedparabolic datum on X.

Let VectDΓ (Y,N) denote the subcategory of VectΓ(Y ) consisting of all rank 2Γ–vector bundles V over Y of fixed local type (see [3, 2.4.1] for details).

A Γ–vector bundle V is called Γ–stable (resp. Γ–semistable) iff for all Γ–invariant line subbundles L ⊂ V the following holds

(2.10) c1(L) · H < (resp. ≤)c1(V ) · H

2.

Note that the above definition of Γ–stability is strictly weaker than the usual defini-tion of stability; in particular the notion of Γ– stability does not imply the stability

10

PARABOLIC BUNDLES 5

of the underlying Γ–bundle. In contrast, the notion of Γ–semistability is equiva-lent to the usual notion of semistability of the underlying Γ–bundle because of theuniqueness of the Harder-Narasimhan filtration.

2.6.Parabolic bundles and Γ–bundles. In [4] a categorical correspondence

between the objects of PVect(X,D,N) and the objects of VectDΓ (Y,N) has beenconstructed, induced by the “invariant direct image” functor pΓ∗ . The details of thisidentification is also given in [2, Section 2].

Let H denote the pullback p∗(H). Then the above correspondence betweenparabolic bundles on X and Γ–bundles on Y identifies the Γ–semistable (resp. Γ–stable) objects with parabolic semistable (resp. parabolic stable) objects as well.The invariant direct image functor pΓ∗ giving this equivalence of categories is more-over a “tensor functor” which sends the usual dual of a Γ–vector bundle to the“parabolic dual” of the corresponding parabolic vector bundle .

2.7. Γ–derived functors Let C be a C-linear abelian category with enoughinjectives. Let Γ be a finite group. Let CΓ be the category whose objects arepairs of the form (A, ρ : Γ → AutC(A)) where A ∈ C. A morphism betweenpairs (A, ρ : Γ → AutC(A)), (B, ρ′ : Γ → AutC(B)) is defined as a Γ–equivariantmorphism in C, i.e. the diagram

(2.11) Aρ(γ) ��

f

��

A

f

��B

ρ′(γ) �� B

is required to commute for all γ ∈ Γ.Since the ground field is assumed to be of characteristic 0, for any object

(A, ρ) ∈ CΓ, we have a subobject AΓρ ⊂ A defined as follows. Given γ ∈ Γ and

A ∈ C, we can define the γ-invariant subobject Aγ of A to be the kernel of thecomposite map:

AΔ−→ A⊕A

id⊕(−ρ(γ))−→ A

where Δ is the diagonal morphism. We define the Γ-invariant subobject AΓρ of A

to be the intersection of the Aγ ’s in A, γ ∈ Γ, i.e.

(2.12) AΓρ :=

⋂γ∈Γ

Ker((id⊕ (−ρ(γ))) ◦Δ).

Note that the induced action of Γ on AΓρ is trivial. Therefore we will just write AΓ

instead of AΓρ .

Let F be a covariant left exact functor from C to an abelian category B. Anyk-linear additive functor F : C → D extends uniquely to a functor F : CΓ → DΓ

since any action of Γ on an object A extends to an action of Γ on F (A). Let FΓ

be the invariant functor which sends A to F (A)Γ. It is a subfunctor of F . We havethe following useful observation.

Lemma 2.8. FΓ is a direct summand of F . Consequently the right derivedfunctors RiFΓ are direct summands of RiF .

Proof. The fact that FΓ is a direct summand of F follows immediately fromthe assumption on the characteristic of the ground field. �

11


Now we return to the case in which we are interested. Let Y be a Kawamatacover of X with a finite group Γ acting on Y such that Y/Γ = X. Note that by ourprevious notations if C denotes the category Vect(Y ) of vector bundles on Y then

CΓ is the category VectΓ(Y ) of Γ-vector bundles on Y . The global section functorHom(Y,−) gives rise to a left exact functor from CΓ to category of k-linear spaces.We denote the i-th right derived functor of this by Exti(Y,−). Let Exti

Γ(Y,−) be

the right derived functor of the global invariant section functor HomΓ(Y,−).By Lemma 2.8 we have the following proposition.

Proposition 2.9. ExtiΓ(Y,−) is a direct summand of Exti(Y,−). Hence

extiΓ(Y, F ) ≤ exti(Y, F )

for all F ∈ CΓ, where “ext” denotes the dimension of the vector space “Ext”.

Let us consider the category C• of filtered OY -modules whose objects are de-noted by F•, i.e. sheaves F with a filtration of subsheaves

(2.13) F• : 0 = F0 ⊂ F1 · · · ⊂ Fn = F.

Let CΓ,−• be the category whose objects are given by Γ-filtered sheaves of OY mod-

ules (as in (2.13)). For any two objects F•, G• in CΓ,−• , morphisms in CΓ,−

• aredefined as:

HomΓ,−(F•, G•) = {φ : F → G : φ(Fi) ⊂ Gi for all 0 ≤ i ≤ n}.

Let CΓ,+• be the category whose objects are the same as in CΓ,−

• , and morphismsbetween two objects F•, G• are defined as

HomΓ,+(F•, G•) = HomC(F,G)/HomΓ,−(F•, G•).

Both these categories CΓ,±• are abelian categories with enough injectives and

HomΓ,±(F•,−) are both left exact covariant functors. Let Exti

Γ,±(F•,−) be the

right derived functors of HomΓ,±(F•,−).

We have a long exact sequence (cf. [9, page 49])

(2.14) · · · �� ExtiΓ,−(F•, G•) �� Exti

Γ(F•, G•) �� Exti

Γ,+(F•, G•) �� · · ·

3. Rssα is irreducible for small α

3.1. A description of Rssα . We briefly recall the construction of semistable

sheaves over X. For details see ([9, Chapter 4]). Let H = OX(−m)p for some mand p. Let

(3.1) Q := Quot(H, P )

be the Quot scheme which parametrizes quotients of H with fixed Hilbert polyno-mial P given by

(3.2) P (n) := n2H2 + n(c1 ·H −KX ·H) +c21 − c1 ·KX

2− c2 + 2χ(OX),

where KX is the canonical line bundle and the ci are the Chern classes of thesheaves which we wish to parametrize.

For fixed Chern classes c1 and c2, it is known that rank 2 semistable sheaves Fwith ci(F ) = ci can be realized as quotients of a fixed H = OX(−m)p for suitablychosen m and p. Let Rss ⊂ Q (resp. Rs ⊂ Q) consist of points [H → F ] ∈ Qsuch that the quotients F are semi-stable (resp. stable) torsion-free sheaves and the

12

PARABOLIC BUNDLES 7

quotient mapH → F induces an isomorphism Cp = H0(X,H(m)) ∼= H0(X,F (m)).Let Rs ⊂ Rs (resp Rss ⊂ Rss) denote the open subschemes parametrizing locallyfree sheaves.

In [13] the Donaldson-Uhlenbeck moduli space has been constructed as theclosure of the moduli space Ms of the moduli space of stable locally free sheaves inthe scheme Mss together with the reduced scheme structure. The scheme Mss isrealized as the image of a PGL(m)-invariant mapping

π : Rss → Mss

Furthermore, the scheme Mss is projective. Note that Mss is not a GIT quotientbut it maps Rs to an open subset Ms ⊂ Mss and π |Rs is the GIT quotientRs//PGL(m). The closure MDU of Ms in Mss with the reduced scheme structureis the precise algebro-geometric analogue of the differential geometric constructiondue to Donaldson. The key property of the moduli space MDU is that the boundaryof Ms is describable in terms of locally free polystable sheaves with lower c2 andcertain zero cycles.

Lemma 3.2. Rss is irreducible for large c2, for a fixed c1.

Proof. Observe thatRs is a dense open subset inRss for large c2 ([9, Theorem9.1.2, page 200]). So irreducibility of Rss is equivalent to the irreducibility of Rs.Now Ms is a geometric quotient of Rs for the action of PGL(m). By [9, Theorem9.4.3, page 203] the scheme Ms is irreducible for large c2. Since the quotient mapf : Rs → Ms is an open map with both base and fibre being irreducible, it followsthat Rs is irreducible (see Lemma 3.3 below) . �

Lemma 3.3. If f : X → Y is a morphism of schemes such that f is an opensurjective morphism and each closed fibre is irreducible, then

Y irreducible =⇒ X irreducible .

Proof. Let U and V be two nonempty open sets in X. Since f is an opensurjective map and Y is irreducible, f(U) and f(V ) are open nonempty subsets ofY such that f(U) ∩ f(V ) �= ∅. Let y ∈ f(U) ∩ f(V ) be a closed point and x1 ∈ Uand x2 ∈ V such that y = f(x1) = f(x2). Clearly, U ∩ f−1(y) and V ∩ f−1(y) aretwo nonempty open subsets of f−1(y). Since f−1(y) is irreducible it implies thatU ∩ V ⊃ U ∩ V ∩ f−1(y) �= ∅. Hence Y is irreducible. �

3.4. The small weight case. Since our final aim in this paper is to show thatthe Donaldson-Uhlenbeck spaces constructed in [3] are asymptotically irreducible,we will assume for the rest of the paper that the sheaves that we consider in theQuot scheme are locally free. Recall that the scheme Rss (resp. Rs) parametrizessemistable (resp. stable) locally free quotients. We will stick to these assumptionsand notations in the paper from here onwards.

We now consider bundles equipped with parabolic structures. The weightα : = (α1, α2) is called small if it satisfies the condition

(3.3) (α2 − α1)D ·H <1

2.

Now a key observation, which is easy to check, is the following.

Lemma 3.5. For small weights α (3.3), for any E ∈ Rs and any quasi-parabolicstructure (2.4), the parabolic bundles E∗ is α-stable and conversely any α-stableparabolic bundle E∗ has the property that its underlying bundle E is semistable.

13


Let T be the total family parametrizing quasi-parabolic structures on rank 2bundles. The notation T is loose since it hides the topological and parabolic data ofits underlying objects. However, we observe that T is independent of the parabolicweights α, β etc.

Since we do not need to make modifications in the topological datum to proveasymptotic irreducibility we do not carry it as a part of the notation. Again since therank of the bundle is 2 there is not much in terms of the quasi-parabolic structureexcept the degree of the subbundle when restricted to the parabolic divisor D. Thiswill figure in the discussion that follows. It will be mentioned whenever needed andshould cause no confusion.

Let Rαk,d be the total family for H–stable parabolic bundles with weight α. For

a formal definition of Rαk,d we direct the reader to [15]. We simplify the notation

and haveRs

α := Rαk,d.

By the definition of T we have the obvious morphism, namely forget : T → Rwhich “forgets” the quasi-parabolic structure. Note however that under this mapthe image of Rss

β or Rsβ need not be contained in Rss; similarly, the inverse image

of Rs can fall outside Rsβ for an arbitrary weight β.

In our simple setting of a flag which is only one-step on a rank 2 bundle, whenthe weights are small (3.3), the morphism forget : Rs

α → Rss is well-defined and theinverse image of Rs is contained in Rs

α (this is a consequence of Lemma 3.5). Let

T s and T ss denote the inverse images of Rs and Rss in T . In other words, T ss isthe total family of quasi-parabolic structures on semistable bundles.

The upshot is that, if the weight α is small we have open inclusions:

T s ⊂ Rsα ⊂ T ss.(3.4)

We now describe the space T ss of quasi-parabolic structures on rank 2 semistablebundles. Let F be the universal sheaf on X × Rss and let L be the Poincare linebundle over D × Picl(D). We have a diagram of various projections:

(3.5) D ×Rss × Picl(D)

p1

��p2

��p3

��

D ×Rss Rss × Picl(D) D × Picl(D).

LetW := p2∗(Hom(p∗1(F |D×Rss), p∗3(L))),

where we have assumed that l = deg(Lt) is sufficiently large so that

p2∗(Hom(p∗1(F |D×Rss), p∗3(L)))is locally free ([8, page 288]). Let

Z = Spec Sym(W∗)

be the underlying geometric vector bundle. This scheme parametrizes all morphismsfrom (F |D) → L for F ∈ Rss and L ∈ Picl(D). Let

Zsur ⊂ Z

be the open subscheme which parametrizes the surjective morphisms. It is nothard to show that by choosing l � 0 we can have a non-empty set of surjectivemorphisms (see for example [1, Theorem 2, page 426]). By taking kernels of these

14

PARABOLIC BUNDLES 9

morphisms which give line subbundles, we get the quasi-parabolic structures. Thus,after choosing l suitably, we see that Zsur ⊂ Z is a non-empty open subset. Bythe definition of T ss (for suitable l) it is immediate that there is an isomorphism

Zsur∼−→ T ss.

By Lemma 3.2 the scheme Rss is asymptotically irreducible, being open in Rss;furthermore, Picl(D) is also irreducible for any integer l. Hence Z and therefore Zsur

is asymptotically irreducible. This implies that T ss is asymptotically irreducible. Itfollows by (3.4), that Rs

α is asymptotically irreducible.Observe that bounds on l in turn give bounds on d = c1(F2

D(F )).D, where thequasi-parabolic structure F2

D(F ) ⊂ F |D is obtained as the kernel to F |D→ L. Weisolate this key result in the following proposition.

Proposition 3.6. For small α, Rsα is asymptotically irreducible for suitable

d = c1(F2D(E)).D.

Remark 3.7. The bounds on l, which in turn give bounds on d, ensure that,irrespective of the weight α, the bundles E have enough quasi-parabolic structureson the given divisor D ⊂ X.

4. The density of Msα in Mss

α

Let Msα = Mα

k,d be the moduli stack of α-stable bundles and Mssα the moduli

stack of α-semistable bundles on X with topological and parabolic datum as in§2. The aim of this section is to prove that the open substack Ms

α in the modulistack Mss

α is dense for any α. We handle the problem by converting it to theequivariant Γ-bundle setting. The general set-up is as in §2 and we use the samenotation. Let Y be a Kawamata cover of X. The advantage in doing this is thatthe technical complications arising in handling obstruction theory in the parabolicsetting is considerably simplified when we make this shift.

Let H be a Γ-sheaf over Y and P1, P2 are two fixed polynomials. LetDrapΓ(H, P1, P2) denote the “generalized flag scheme” which parametrizes Γ–sub-sheaves of H

H∗ := 0 ⊂ H3 ⊂ H2 ⊂ H1 = Hsuch that the Hilbert polynomial of Hi−1/Hi is Pi−1. These can be defined asΓ-fixed points of the usual Drap scheme (cf. [3, page 15], [9, Appendix 2.A, page48]).

Lemma 4.1. The dimension of DrapΓ(H, P1, · · · , Pk) at the point H∗ satisfiesthe following inequality

ext0Γ,+

(H,H) ≥ dimH∗(DrapΓ(H, P1, · · · , Pk)) ≥ ext0Γ,+

(H,H)− ext1Γ,+

(H,H).

Proof. The proof of this lemma is a routine equivariant generalization of theone given in [9, Proposition 2.A.12, page 54] and we omit the details. �

4.2. Parabolic Chern classes Let E∗ be a rank 2 parabolic vector bundleover X with underlying bundle E. The parabolic Chern classes are defined as (see[3, Lemma 6.1])

(4.1) par(c1)(E∗) = c1(E) + (α1 + α2) ·D,

(4.2) par(c2)(E∗) = c2(E) + (α1 + α2)c1(E) ·D + α1α2D2.

15


Observe that when c1(E) = 0 (which is our base assumption), both par(c1) andpar(c2) differ from the usual c1 and c2 by terms which involve only the parabolicdivisor.

Let V be a Γ–bundle on Y . We have (see [4, Equation 3.11])

(4.3) c1(V ) = p∗(par(c1)({pΓ∗ (V )}∗)),

where {pΓ∗ (V )}∗ is the invariant direct image of V with the canonical parabolicstructure coming from the Seshadri-Biswas correspondence.

For a Γ-bundle F of rank 2 of local type τ (α) (see [3, Definition 2.12] for thedefinition) we have the equation in the second Chern classes of the underlyingbundle:

(4.4) c2(pΓ∗ (Hom(F, F ))) = c2

[{pΓ∗ (F )}∨∗ ⊗{pΓ∗ (F )}∗

]= 4 c2((p

Γ∗ (F ))),

where the last equality follows by a splitting principle argument as in [3, Lemma6.1] and the assumption that c1((p

Γ∗ (F )) = 0.

Here and elsewhere, “∨” denotes the parabolic dual and ⊗ denotes the parabolictensor product. By the naturality of parabolic Chern classes we have

(4.5) par(ci)(E∨∗ ) = (−1)ipar(ci)(E∗).

When we work with a Kawamata cover as in our case, then we have the followingrelation between the Γ–cohomology and the usual cohomology on Y/Γ = X:

(4.6) HiΓ(Y,F) = Hi(X, pΓ∗ (F)), ∀i.

Definition 4.3. For Γ-bundles F and G on Y , define

χΓ(F,G) :=

∑i

(−1)iextiΓ(F,G).

Let V be a Γ-bundle of rank r on Y . Define the Γ-discriminant of V as:

ΔΓ(V ) := 2r c2(p

Γ∗ (V ))− (r − 1) c1(p

Γ∗ (V ))2.

4.4. Γ-total families. Let RssΓ

(resp. RsΓ) parametrize Γ–semistable (resp.

stable) bundles of type τ (α) and fixed topological datum (c1, c2) over Y . In §3 and§4 of [3] we give the construction of Rss

Γwhich parametrizes Γ-torsion–free sheaves.

We recall that there is an action of Γ on a suitable Quot scheme of quotients on theKawamata cover Y of X. The scheme R

Γis the subscheme of Γ–fixed points in the

Quot scheme which consists of torsion–free sheaves and RssΓ

is an open subscheme ofR

Γ. We stick to locally free sheaves in this work since we work with the Donaldson-

Uhlenbeck compactifications.

4.5. Cohomological computations. The following lemmas play a key rolein proving that Rs

Γis dense in Rss

Γ.

Lemma 4.6. Let F be a Γ–vector bundle of rank 2 on Y of some type τ (α),such that

(4.7) c1(pΓ∗ (F )) = 0 .

Then

χΓ(F, F ) = −ΔΓ(F ) + 4χ(OX).

16

PARABOLIC BUNDLES 11

Proof. We have χΓ(F, F ) = χΓ(Y,Hom(F, F ))

= χ(X, pΓ∗ (Hom(F, F ))) (because of (4.6))= −c2(p

Γ∗ (Hom(F, F ))) + 4χ(OX) (by Hirzebruch-Riemann-Roch and (4.7))

= −4 c2(pΓ∗ (F )) + 4χ(OX) (by (4.4))

= −ΔΓ(F ) + 4χ(OX) (by Definition 4.3). �Let F ∈ Rss

Γ− Rs

Γ be a rank 2 strictly Γ-semistable bundle on Y such that

c1(pΓ∗ (F )) = 0. Let

0 → F1 → F → F2 → 0

be the Γ-Jordan-Holder filtration of F . Observe that the F2 is a torsion–free Γ–sheaves of rank 1, while F1 is locally free.

Let pΓ∗ (F ) = E∗ and pΓ∗ (Fi) = Ei,∗, i = 1, 2. Then

0 → E1,∗ → E∗ → E2,∗ → 0

is the parabolic Jordan-Holder filtration of E∗ on X. Note that E2,∗ is a parabolictorsion-free sheaf of rank 1. Further, c1(E) = c1(p

Γ∗ (F )) = 0. We remark that if the

parabolic line bundle E1,∗ has weight α1, then its parabolic dual E∨1,∗ has weight

1 − α1 (cf. [12, Section 8] where the weight will be simply −αi in the balancedconvention).

Lemma 4.7. Let Ei,∗ be as above with weights αi on X. Then

χ(X,E2,∗⊗E∨1,∗) = χ(X,E2 ⊗ E∗

1 )

E∗1 being the usual dual of E1.

Proof. By the Hirzebruch-Riemann-Roch theorem (K being the canonicaldivisor on X), we see that

χ(X,E2,∗⊗E∨1,∗) =

c1(E2,∗⊗E∨1,∗)

2

2− c1(E2,∗⊗E∨

1,∗) ·K2

+ χ(OX).

We write c1(E2,∗⊗E∨1,∗) for the Chern class of the underlying bundle (and not its

parabolic Chern class) since it is notationally inconvenient to shed the parabolicluggage on the tensor product E2,∗⊗E∨

1,∗, the reason being that the underlying

sheaf of E2,∗⊗E∨1,∗ is not E2 ⊗ E∗

1 .Observe thatc1(E2,∗⊗E∨

1,∗) = par(c1)(E2,∗⊗E∨1,∗)− (α2 − α1)D

= par(c1)(E2,∗) + par(c1)(E∨1,∗)− (α2 + 1− α1)D

= [par(c1)(E2,∗)− α2D] + [par(c1)(E∨1,∗)− (1− α1)D]

= c1(E2) + c1(E1∗) = c1(E2 ⊗ E∗

1 )

and the result follows. �Let

ξ21 = c1(E2)− c1(E1).

Observe that, by the definition of ΔΓ , we have

(4.8) ΔΓ(F ) = 4c2(E) = 4c1(E1) · c1(E2).

Now

(4.9) [c1(E2)− c1(E1)]2 = [c1(E2) + c1(E1)]

2 − 4[c1(E2) · c1(E1)]

= −4[c1(E2) · c1(E1)]

since c1(E2) + c1(E1) = c1(E) = 0.Hence by (4.8) and (4.9)

(4.10) ξ221 = [c1(E2)− c1(E1)]2 = −ΔΓ(F ).

17


The Γ-Euler characteristic has the following description for the Γ-line bundlesFi.

(4.11) χΓ(F1, F2) = χ(X, pΓ∗ (Hom(F1, F2))) = χ(X, pΓ∗ (F2 ⊗ F ∗1 ))

= χ(X,E2,∗⊗E∨1,∗) =

ξ2212

− ξ21.K

2+ χ(OX)

by the proof of Lemma 4.7. Hence

(4.12) χΓ(F1, F2) =ξ2212

− ξ21.K

2+ χ(OX).

Lemma 4.8. Let F ∈ RssΓ

−RsΓ be as above such that c1(p

Γ∗ (F )) = 0. Then

ext1Γ,−(F, F ) ≤ 3

4ΔΓ(F ) +B

where B is an irrelevant number not involving the Chern classes of the bundles.

Proof. By a Γ-equivariant version of the spectral sequence in [9, 2.A.4] wehave

ext1Γ,−(F, F ) ≤ ext1

Γ(F1, F1) + ext1

Γ(F2, F1) + ext1

Γ(F2, F2)

= {ext0Γ(F1, F1) + ext2

Γ(F1, F1)− χΓ(F1, F1)} +{ext0

Γ(F2, F1) + ext2

Γ(F2, F1)−

χΓ(F2, F1)} +{ext0Γ(F2, F2) + ext2

Γ(F2, F2)− χΓ(F2, F2)}

≤ B1 − {χΓ(F1, F1) + χΓ(F2, F1) + χΓ(F2, F2)} (1)= B1 + χΓ(F1, F2)− χΓ(F, F )

= B1 +ξ2212

− ξ21.K2

+ΔΓ(F )− 3χ(OX) (by Lemma 4.6 and (4.12))

= 34ΔΓ(F ) +

ξ2214

− ξ21.K2

+B1 − 3χ(OX) (by (4.10))

= 34ΔΓ(F ) +

[ξ212

− K2

]2+B1 − 3χ(OX)− K2

4

≤ 34ΔΓ(F ) +B.

The last inequality with the irrelevant number B comes by the following rea-soning. By the Hodge index theorem,

[ξ212

− K

2

]2≤

([ξ212

− K2

]·H

)2H2

.

Further, by the parabolic semistability of E∗, since Ei,∗ are its parabolic Jordan-Holder terms we have

par(c1)(E1,∗) ·H =par(c1)(E∗) ·H

2= par(c1)(E2,∗) ·H.

Hence, (c1(E2) − c1(E1)) ·H = ξ21 ·H = (α1 − α2)D ·H is an irrelevant number.

The remaining terms in ([ξ212 − K

2

]·H) are clearly irrelevant. �

4.9. The density of RsΓin Rss

Γ. In the rest of this section we conclude the

density of RsΓin Rss

Γ.

Lemma 4.10. There is an irrelevant number B depending on the rank, X, H,the parabolic datum αi and D, such that

dim(RssΓ

−RsΓ) ≤ endΓ(H) +

3

4ΔΓ(F ) +B

where F ∈ RssΓ

−RsΓ

1Since Fi are rank 1 Γ-torsion–free sheaves, the dimensions ext0Γ(Fi, Fj) and ext2Γ(Fi, Fj) are

bounded ∀i, j and the irrelevant number B1 is to take care of these terms.

18


Proof. Let {ρ : H → F} ∈ RssΓ

−RsΓwith Γ–Jordan-Holder filtration

0 = F0 ⊂ F1 ⊂ F2 = F(4.13)

such that F1 and F2/F1 are rank 1 torsion–free sheaves with the same μ = μ(F )and F1 is locally free. The filtration 4.13 induces a Γ-filtration on H

0 ⊂ H0 ⊂ H1 ⊂ H2 = H(4.14)

such that F1 = H1/H0 and F2 = H2/H0.Let P∗ := (P1, P2) be the Hilbert polynomials of H2/H1 and H1/H0. Since we

fix the topological type of these quotients, we get a bounded family of sheaves withfixed μ = μ(F ) giving only finitely many choices of P∗. Let Z be the finite unionof DrapΓ(H, P1, P2). There is a morphism f : Z → QΓ (the Γ–fixed points of theQuot scheme on Y ) sending H0 ⊂ H1 ⊂ H to H0 ⊂ H. It is clear that Rss

Γ−Rs

Γ⊂

f(Z) (since every strictly semistable object has a Jordan-Holder filtration).We have by Lemma 4.1

(4.15) dim(RssΓ

−RsΓ) ≤ dimZ ≤ ext0Γ,+(H,H).

The definition of Ext± gives an exact sequence

0 → Ext0Γ,−(H,H) → Ext0Γ(H,H) → Ext0Γ,+(H,H) → Ext1Γ,−(H,H).

Hence

ext0Γ,+(H,H) ≤ endΓ(H)− ext0Γ,−(H,H) + ext1Γ,−(H,H)

≤ endΓ(H)− 1 + ext1Γ,−(F ,F).

The last inequality follows from the fact that a filtration of F canonicallyinduces a filtration on H, and we also have ext1Γ,−(F ,F) = ext1Γ,−(H,H). Hence,

ext0Γ,+(H,H) ≤ endΓ(H) +3

4ΔΓ(F ) +B (by Lemma 4.8).

�Proposition 4.11. For any {ρ : H → F} ∈ Rss

Γ ,

dimρ(RssΓ) ≥ endΓ(H) + ΔΓ(F )− 4χ(OX)

Proof. We follow the proof of O’Grady (see [16] and [9, Theorem. 4.5.8,page. 104]. Let K be the kernel of the morphism ρ. Applying HomΓ(−, F ) to

0 −→ K −→ H −→ F −→ 0,

we get

0 −→ EndΓ(F ) −→ HomΓ(H, F ) −→ HomΓ(K,F ) −→ Ext1Γ(F, F ) −→ 0

Suppose that a positive integer m has been already chosen for which F is m–regular. Therefore, as we have seen earlier, H0(H(m)) = H0(F (m)). Thus wehave

HomΓ(H, F ) = HomΓ(H0(H(m)), H0(F (m))) = HomΓ(H,H)

and we have the following equality of dimensions:

homΓ(K,F ) = {endΓ(H)− endΓ(F ) + ext1Γ(F, F )}.Using this computation we get the following inequality of dimensions:

dimρ(RssΓ ) ≥ homΓ(K,F )− ext2Γ(F, F )

= endΓ(H)− endΓ(F ) + ext1Γ(F, F )− ext2Γ(F, F ) = endΓ(H)− χΓ(F, F )

= endΓ(H) + ΔΓ(F )− 4χ(OX) (by Lemma 4.6). �

19


Proposition 4.12. Let B be as in Lemma 4.10. If ΔΓ(F ) > 4(B + 4χ(OX)),

then RsΓis dense in Rss

Γ.

Proof. We have the following inequalities:dim(Rss

Γ−Rs

Γ) ≤ {endΓ(H) + 3

4ΔΓ(F ) +B}

< {endΓ(H) + ΔΓ(F )− 4χ(OX)} ≤ min{dim(Xi) : Xi a component of RssΓ}.

Note that the above inequalities show that the dimension of any component isat least (endΓ(H)+ΔΓ(F )− 4χ(OX)). Hence Rs

Γintersects all components of Rss

Γ.

Hence RsΓis dense in Rss

Γ. �

Corollary 4.13. If the second Chern class c2 of the underlying bundles islarge, then Ms

α is dense in Mssα for any weight α.

Proof. By the Seshadri–Biswas correspondence we see that the moduli stackMss

Γ of Γ–semistable bundles (resp. MsΓ of Γ–stable bundles) is isomorphic to Mss

α

(resp. Msα). Hence by Proposition 4.12, since c2 � 0 it follows that Rs

Γis dense in

RssΓ. Since Rss

Γ(resp. Rs

Γ) is the atlas of the Artin stack Mss

Γ (resp. MsΓ) we have

a surjective morphism RssΓ

→ MssΓ and the result follows. �

5. Variation of Mssα and the Main Theorem

Let Mssα (resp. Ms

α) be the moduli stack of α-semistable (resp. α-stable)bundles with first Chern class 0. Here we tacitly assume that d is chosen as inProposition 3.6 (see Remark 3.7). Now we study these moduli stacks as we varyweights.

Let N0 = D ·H,

W = {(α1, α2) : 0 < α1 < α2 < 1}and

δW :=

{(α1, α2) : 0 < α1 < α2 < 1 such that | α1 − α2 |= k

2N0, 1 ≤ k ≤ 2N0

}.

Let W ◦ = W − δW A connected component of W ◦ is called a chamber. Observethat, if α is a weight within a chamber then Mss

α = Msα. Moreover if α and β are

in same chamber then Mssα = Mss

β and also Msα = Ms

β .We have the following lemma:

Lemma 5.1. If α is in a chamber and ω is on an adjacent wall, then anyω-stable bundle is α-stable and any α-stable bundle is ω-semistable.

Proof. For 0 ≤ t ≤ 1, let αt denote the vector tα+(1− t)ω in W . Then, αt isalso in the chamber for t �= 0. Suppose that E∗ is ω-stable and suppose that E∗ isnot α-stable. Then, there exists a subbundle E′

∗ of E∗ such that μα(E′∗) ≥ μα(E∗).

The continuous function t �→ μαt(E′

∗)− μαt(E∗) assumes a negative value at t = 0

and is non-negative at t = 1 and hence takes the value 0 for some 0 < t0 ≤ 1. Butthen E∗ is strictly semistable with respect to the weight αt0 contradicting the factthat αt0 is within the chamber.

Similarly, if E∗ is α-semistable (therefore in fact α-stable) and E∗ is not ω-semistable, then there exists a subbundle E′

∗ of E∗ such that μω(E′∗) > μω(E∗).

Thus, μαt(E∗)−μαt

(E′∗) is negative at t = 0 and non-negative at t = 1; this would

imply that with respect to some αt0 within the chamber, E∗ is strictly semistable,again a contradiction. �

We have the following:

20


Corollary 5.2. If α is in a chamber and ω is in a adjacent wall then we havethe following inclusions:

Msω ⊂ Ms

α = Mssα ⊂ Mss

ω .

Theorem 5.3. Mβ is asymptotically irreducible for all β.

Proof. Recall that the scheme Rsα is an atlas for the Artin stack Ms

α and wehave a canonical surjective morphism Rs

α → Msα. Hence by Proposition 3.6, Ms

α

is asymptotically irreducible for small α.For any ω in an adjacent wall, by Corollary 5.2 we see thatMs

ω is asymptoticallyirreducible being an open substack of Ms

α. Now by Corollary 4.13 it follows thatMss

ω is asymptotically irreducible.Now taking β in any chamber with ω in an adjacent wall and different from

the “small” chamber, we see again by Corollary 4.13 that Mssβ is asymptotically

irreducible. We proceed similarly to reach all weights in W using the connectednessof W and finiteness of the number of walls; since Ms

β surjects onto Msβ it follows

that Msβ is asymptotically irreducible.

Now recall that Mβ is by definition the closure of Msβ (with the reduced scheme

structure) in a certainMβk,d. This implies thatMβ is also asymptotically irreducible

and the theorem follows. �

Remark 5.4. The subtle point is that even though we finally need to provethat Ms

α is asymptotically irreducible, we are forced to go to the semistable bundlessince we need to go over various weight chambers.

Remark 5.5. Observe that the arguments in this paper automatically give asa consequence the generic smoothness and asymptotic non-emptiness of the modulispaceMα. In specific situations, one can use the techniques of this paper to concluderationality of certain parabolic moduli.

References

[1] M. F. Atiyah, Vector bundles over an elliptic curve, Proc. London Math. Soc. (3) 7 (1957),414–452.

[2] V. Balaji, I. Biswas, D. S. Nagaraj, Principal bundles over projective manifolds with parabolicstructure over a divisor, Tohoku Math. J. (2) 53 (2001), no. 3, 337–367.

[3] V. Balaji, A. Dey, R. Parthasarathi, Parabolic bundles on algebraic surfaces, I. TheDonaldson-Uhlenbeck compactification, Proc. Indian Acad. Sci. Math. Sci. 118 (2008), no. 1,43–79.

[4] I. Biswas, Parabolic bundles as orbifold bundles, Duke Math. J. 88 (1997), no. 2, 305–325.[5] H. U. Boden, Y. Hu, Variations of moduli of parabolic bundles, Math. Ann. 301 (1995), no.

3, 539–559.[6] D. Gieseker, J. Li, Irreducibility of moduli of rank-2 vector bundles on algebraic surfaces, J.

Differential Geom. 40 (1994), no.1, 23–104.[7] A. Grothendieck, Sur quelques points d’algebre homologique, Tohoku Math. J. (2) 9 (1957),

119–221.[8] R. Hartshorne. Algebraic Geometry. Graduate Texts in Mathematics 52. Springer-Verlag,

New York-Heidelberg, 1977.[9] D. Huybrechts, M. Lehn. The geometry of moduli spaces of sheaves. Aspects of Mathematics,

E31. Friedr. Vieweg und Sohn, Braunschweig, 1997.[10] Y. Kawamata, Characterization of abelian varieties, Compositio Math. 43 (1981), 253–276.[11] Y. Kawamata, K. Matsuda, K. Matsuki, Introduction to the minimal model problem, Alge-

braic Geometry, Sendai, 1985, 283–360, Adv. Stud. Pure. Math. 10, North-Holland, Amster-dam, 1987.

21


[12] P. B. Kronheimer, T. S. Mrowka, Gauge theory for embedded surfaces II, Topology, 34 (1995),No. 1, 37–97.

[13] J. Li, Algebraic geometric interpretation of Donaldson’s polynomial invariants, J. DifferentialGeom. 37 (1993), no.2, 417–466.

[14] M. Maruyama, Instantons and parabolic sheaves, Geometry and analysis (Bombay, 1992),245–267, Tata Inst. Fund. Res., Bombay, 1995.

[15] M. Maruyama, K. Yokogawa, Moduli of parabolic stable sheaves, Math. Ann. 293 (1992),

no.1, 77–99.[16] K. G. O’Grady, Moduli of vector bundles on projective surfaces: some basic results, Invent.

Math. 123 (1996), no.1, 141–207.[17] C. S. Seshadri, Moduli of π-vector bundles over an algebraic curve, 1970 Questions on Alge-

braic Varieties (C.I.M.E, III Ciclo, Varenna, 1969), pp. 139–260. Edizione Cremonese, Rome.

Chennai Mathematical Institute, Plot H1, SIPCOT IT Park Padur PO, Siruseri

603103, India

E-mail address: [email protected]

School of Mathematics, Tata Institute of Fundamental Research, Homi Bhabha

Road, Bombay 400005, India


22


Finite Subgroups of PGL2(K)

Arnaud Beauville

To Ramanan on his 70th birthday

Abstract. We classify, up to conjugacy, the finite subgroups of PGL2(K) oforder prime to char(K).

Introduction

The aim of this note is to describe, up to conjugacy, the finite subgroups ofPGL2(K), for an arbitrary field K. Throughout the paper, we consider only sub-groups whose order is prime to the characteristic of K.

When K = C, or more generally when K is algebraically closed, the answeris well known: any such group is isomorphic to Z/r, Dr (the dihedral group), A4,S4 or A5, and there is only one conjugacy class for each of these groups. If Kis arbitrary, the group PGL2(K) is contained in PGL2(K), so the subgroups ofPGL2(K) are among the previous list; it is not difficult to decide which subgroupsoccur for a given field K, see §1.

So the only question left is to describe the conjugacy classes in PGL2(K) of thesubgroups in the list. In §2 we give a general answer for subgroups of G(K), for analgebraic group G, in terms of (non-abelian) Galois cohomology. We illustrate themethod on one example in §3, and apply it to the case G = PGL2 in §4.

The motivation for looking at this question was to understand the appearanceof the Brauer group in the case of (Z/2)2 considered in [B]. The result is somewhatdisappointing, as it turns out that this case (which could be treated directly, as in[B]) is the only one where some second Galois cohomology group plays a role. Atleast our method explains this role, and hopefully may be useful in other situations.

1. The possible subgroups

We repeat that whenever we mention a finite group, we always assume that itsorder is prime to the characteristic of K. The following is classical (see [S2], 2.5).

Proposition 1.1. 1) PGL2(K) contains Z/r and Dr1 if and only if K contains

ζ + ζ−1 for some primitive r-th root of unity ζ.

2010 Mathematics Subject Classification. 20G15.1We denote by Dr the dihedral group with 2r elements.


1



23

2 ARNAUD BEAUVILLE

2) PGL2(K) contains A4 and S4 if and only −1 is the sum of two squares inK.

3) PGL2(K) contains A5 if and only if −1 is the sum of two squares and 5 isa square in K.

Proof. One way to prove this is to use the isomorphism PGL2(K) ∼−→SO(K, q), where q is the quadratic form q(x, y, z) = x2 + yz on K3 ([D], II.9).If a group H embeds into SO(K, q), we have a faithful representation ρ of H in K3,which preserves an indefinite quadratic form.

• Case H = Z/r : let g be a generator; the existence of q forces the eigenvaluesof ρ(g) in K to be of the form (ζ, ζ−1, 1), with ζ a primitive r-th root of 1. Thisimplies ζ + ζ−1 ∈ K. Conversely, if λ := ζ + ζ−1 is in K, the homography z �→(λ+ 1)z − 1

z + 1is an element of order r of PGL2(K).

• Case H = Dr : by the previous case, if Dr ⊂ PGL2(K), λ := ζ+ζ−1 is in K.

Conversely if λ ∈ K, the homographies z �→ 1/z and z �→ (λ+ 1)z − 1

z + 1generate a

subgroup of PGL2(K) isomorphic to Dr.

• Cases H = A4,S4 or A5 . The representation ρ must be irreducible. Eachof the groups A4 and S4 has exactly one irreducible 3-dimensional representationwith trivial determinant, which is defined over the prime field; the only invariantquadratic form (up to a scalar) is the standard form q0(x, y, z) = x2+y2+z2. ThusA4 and S4 are contained in PGL2(K) if and only if q0 is equivalent to λq for someλ ∈ K∗, which means that q0 represents 0.

Since A5 contains elements of order 5, the condition√5 ∈ K is necessary.

Suppose this is the case, and put ϕ = 12 (1 +

√5); the subgroup of SO(K, q0)

preserving the icosahedron with vertices

{(±1, 0,±ϕ) , (±ϕ,±1, 0) , (0,±ϕ,±1)}is isomorphic to A5. It follows as above that A5 embeds in SO(K, q) if and only ifq0 represents 0. �

2. Some Galois cohomology

2.1. In this section we consider an algebraic group G over K, and a subgroupH ⊂ G(K). We choose a separable closure Ks of K, and put g := Gal(Ks/K). Weare interested in the set of embeddings H ↪→ G(K) which are conjugate in G(Ks)to the natural inclusion i : H ↪→ G(K), modulo conjugacy by an element of G(K).We denote this (pointed) set by Embi(H,G(K)).

We will use the standard conventions for non-abelian cohomology, as explainedfor instance in [S3], ch. I, §5. We will also use the notation of [S3] for Galoiscohomology: if G is an algebraic group over K, we put Hi(K,G) := Hi(g, G(Ks)).

Proposition 2.2. Let Z be the centralizer of H in G(Ks). The pointed setEmbi(H,G(K)) is canonically isomorphic to the kernel of the natural mapH1(K,Z) → H1(K,G).

Proof. Let X ⊂ G(Ks) be the subset of elements g such that g−1 σg ∈ Z forall σ ∈ g. The group G(K) (resp. Z) acts on X by left (resp. right) multiplication.By [S3], ch. I, 5.4, cor. 1, the kernel of H1(K,Z) → H1(K,G) is identified with the(left) quotient by G(K) of the subset of g-invariant elements in G(Ks)/Z; but this

24

FINITE SUBGROUPS OF PGL2(K) 3

subset is by definition X/Z, so we can identify our kernel to the double quotientG(K)\X/Z.

For every g ∈ X, the conjugate embedding gig−1 belongs to Embi(H,G(K)).Any element j ∈ Embi(H,G(K)) is of the form gig−1 for some g ∈ G(Ks); forσ ∈ g, the element σg again conjugates i to j, hence g−1 σg ∈ Z and g ∈ X.Thus the map g �→ gig−1 from X to Embi(H,G(K)) is surjective. Two elements gand g′ of X give the same element in Embi(H,G(K)) if and only if g′ belongs tothe double coset G(K)gZ. Therefore the above map induces a canonical bijectionG(K)\X/Z ∼−→ Embi(H,G(K)). �

2.3. Let us write down the correspondence explicitly: a class in our kernel isrepresented by a 1-cocycle g → Z which becomes a coboundary in G, hence is ofthe form σ �→ g−1 σg for some g ∈ X; we associate to this class the embeddinggig−1.

2.4. We are actually more interested in the set Conj(H,G(K)) of subgroupsof G(K) which are conjugate to H in G(Ks), modulo conjugacy by G(K). Associ-ating to an embedding its image defines a surjective map im : Embi(H,G(K)) →Conj(H,G(K)). The normalizer N of H in G(Ks) acts on H by automorphisms,hence also on Embi(H,G(K)). Two embeddings with the same image differ byan automorphism of H, which must be induced by an element of N if the em-beddings are conjugate under G(Ks). It follows that im induces an isomorphismEmbi(H,G(K))/N ∼−→ Conj(H,G(K)).

2.5. Let us translate this in cohomological terms. Let H1(K,Z)0 denote thekernel of the map H1(K,Z) → H1(K,G). An element n of N acts onEmbi(H,G(K)) by j �→ j ◦ int(n−1); if j = gig−1, this amounts to replace gby gn, hence the 1-cocycle ϕ : σ �→ g−1 σg by n−1ϕ σn. This formula defines anaction of N on H1(K,Z) which preserves H1(K,Z)0; the map g �→ gHg−1 inducesan isomorphism of pointed sets H1(K,Z)0/N

∼−→ Conj(H,G(K)).

3. An example

3.1. In this section we fix an integer r ≥ 2, prime to char(K), and we assumethat K contains a primitive r-th root of unity ζ. We consider the matrices A,B ∈Mr(K) defined on the canonical basis (e1, . . . , er) of K

r by

A · ei = ei+1 , B · ei = ζiei

for 1 ≤ i ≤ r, with the convention er+1 = e1.The matrices A and B generate the K-algebra Mr(K), with the relations

Ar = Br = I , BA = ζAB .

Their classes A, B in PGLr(K) commute; we consider the embedding i : (Z/r)2 ↪→PGLr(K) which maps the two basis vectors to A and B. The image H of i is itsown centralizer; in particular, H is a maximal commutative subgroup of PGLr(K).

By the Kummer exact sequence (and the choice of ζ), the group H1(K,Z/r) isidentified with K∗/K∗r; the pointed set H1(K,PGLr) can be viewed as the set ofisomorphism classes of central simple K-algebras of dimension r2 ([S1], X.5).

Lemma 3.2. Let α, β ∈ K∗, and let α, β be their images in K∗/K∗r. The mapH1(i) : H1(K,Z/r)2 → H1(K,PGLr) associates to (α, β) the class of the cyclic

25

4 ARNAUD BEAUVILLE

K-algebra Aα,β generated by two variables x, y with the relations xr = α, yr = β,yx = ζxy.

Proof. We choose α′, β′ in Ks with α′r = α and β′r = β. The Kummerisomorphism associates to (α, β) the homomorphism (a, b) : g → (Z/r)2 defined by

σα′ = ζa(σ)α′ σβ′ = ζb(σ)β′ for each σ ∈ g .

Its image in H1(K,PGLr(Ks)) is the class of the 1-cocycle σ �→ Aa(σ)Bb(σ).Now let us recall how we associate to the algebra Aα,β a cohomology class

[Aα,β ] in H1(K,PGLr) (loc. cit.). We choose an isomorphism of Ks-algebras u :Mr(Ks)

∼−→ Aα,β ⊗K Ks. For each σ ∈ g, u−1 σu is an automorphism of Mr(Ks),hence of the form int(gσ) for some gσ in PGLr(Ks). Then [Aα,β ] is the class of the1-cocycle σ �→ gσ.

In our case we define u on the generators A,B by u(A) = β′y−1, u(B) = α′−1x.Then the automorphism u−1 σu multiplies A by ζb(σ) and B by ζ−a(σ), which givesgσ = Aa(σ)Bb(σ) as above. �

3.3. The exact sequence

1 → Gm → GLr → PGLr → 1

gives rise to a coboundary homomorphism ∂r : H1(K,PGLr) → H2(K,Gm) =Br(K) which is injective (loc. cit.). The class ∂r[Aα,β ] ∈ Br(K) is the symbol(α, β)r; it depends only on the classes of α and β (mod. K∗r). The map ( , )r :(K∗/K∗r)2 → Br(K) is bilinear and alternating. Since ∂r is injective, we find:

Proposition 3.4. The set Embi((Z/r)2,PGLr(K)) is isomorphic to the set of

couples (α, β) in (K∗/K∗r)2 such that (α, β)r = 0. �We will describe the correspondence more explicitely in the case r = 2 in the

next section.

4. Conjugacy classes in PGL2(K)

Proposition 4.1. Assume that K is separably closed. Two finite subgroups ofPGL2(K) which are isomorphic (and of order prime to char(K)) are conjugate.

Proof. Again this is certainly well-known; we give a quick proof for complete-ness. The possible subgroups are those which appear in Proposition 1.1.

An element of order r of PGL2(K) comes from a diagonalizable element ofGL2(K), hence is conjugate to the homothety z �→ ζz for some ζ ∈ μr(K) 2; thus acyclic subgroup of order r of PGL2(K) is conjugate to the group Hr of homothetiesz �→ λz, λ ∈ μr(K).

There is only one group Dr containing Hr, namely the subgroup generated byHr and the involution z �→ 1/z; it follows that all dihedral subgroups of order 2rare conjugate to this subgroup.

For the three remaining groups, we use again the isomorphism PGL2(K) ∼−→SO3(K). The groups A4 and S4 have exactly one irreducible representation ofdimension 3 with trivial determinant, while A5 has two such representations whichdiffer by an outer automorphism: this is elementary in characteristic 0, and thegeneral case follows by [I], ch. 15. Therefore two isomorphic subgroups H andH ′ of SO3(K) of this type are conjugate in GL3(K). The only quadratic forms

2As usual we denote by μr(K) the group of r-th roots of unity in K.

26


preserved by H or H ′ are the multiple of the standard form; thus the element gof GL3(K) which conjugates H to H ′ must satisfy tg g = λI for some λ ∈ K.Replacing g by ±μg, with μ2 = λ−1, we have g ∈ SO3(K), hence our assertion. �

Recall that the determinant induces a homomorphism det : PGL2(K) →K∗/K∗2.

Theorem 4.2. 1) PGL2(K) contains only one conjugacy class of subgroupsisomorphic to Z/r (r > 2), A4, S4 or A5.

2) The conjugacy classes of cyclic subgroups of order 2 of PGL2(K) are param-etrized by K∗/K∗2: to α ∈ K∗ (mod.K∗2) corresponds the involution z �→ α/z.

3) The homomorphism det : PGL2(K) → K∗/K∗2 induces a bijective corre-spondence between:

• conjugacy classes of subgroups of PGL2(K) isomorphic to (Z/2)2;• subgroups G ⊂ K∗/K∗2 of order ≤ 4, such that (−α,−β)2 = 0 for all α, β in

G (see (3.3)).4) Assume that μr(K) has order r. The conjugacy classes of subgroups Dr

of PGL2(K) are parametrized by K∗/K∗2μr(K). The subgroup corresponding toα ∈ K∗ (mod. K∗2μr(K) ) consists of the homographies z �→ ζz and z �→ αη/z,for ζ, η ∈ μr(K).

Proof. Using Proposition 4.1 we can apply the method of §3. We give the listof the subgroups of PGL2(Ks) and their centralizers:

H Z/2 Z/r (r > 2) Z/2× Z/2 Dr (r > 2) A4 S4 A5

Z Gm � Z/2 Gm Z/2× Z/2 Z/2 1 1 1

In case 1), we have H1(K,Z) = {1} (using H1(K,Gm) = {1}). The resultfollows from (2.5).

Case 2): This is the case where a direct approach is definitely simpler than ourmethod, so we follow the former and leave the latter to the reader. Let s be aninvolution of PGL2(K), and let α ∈ K∗ such that α ≡ −det(s) (mod. K∗2). Thens is represented by a matrix A ∈ GL2(K) with A2 = α I. In a basis (v,Av) of K2,

we have A =

(0 α1 0

), hence s is conjugate to the involution z �→ α/z. This implies

2).Case 3): Let i : (Z/2)2 ↪→ PGL2(K) be the embedding which maps the basis

vectors e1 and e2 to the involutions z �→ 1/z and z �→ −z. By Proposition 3.4 theset Embi((Z/2)

2,PGL2(K)) is canonically identified to the set of couples (α, β) in(K∗/K∗2)2 with (α, β)2 = 0.

We make the correspondence explicit following (2.3). Let α, β ∈ K∗ with(α, β)2 = 0. This means that the conic x2 − αy2 − βz2 = 0 is isomorphic to P1

K ,thus there exists λ, μ in K with λ2 − α − βμ2 = 0. We choose α′ and β′ in Ks

such that α′2 = α and β′2 = β; as above we define the homomorphisms a andb : g → Z/2 by

σα′ = (−1)a(σ)α′ and σβ′ = (−1)b(σ)β′ for each σ ∈ g .

27

6 ARNAUD BEAUVILLE

Put θ :=β′μ

λ+ α′ =λ− α′

β′μ; let g ∈ PGL2(Ks) be the homography z �→ α′ z − θ

z + θ.

An easy computation gives

g−1 σg = i(a(σ), b(σ)) .

Thus the embedding of (Z/2)2 associated to (α, β) is gig−1; it maps e1 to the

homography h1 : z �→ λu− α

z − λ, and e2 to h2 : z �→ α/z . Note that det(h1) = −β

and det(h2) = −α.Now we have to take into account the action of the normalizer N of H in

PGL2(Ks). This is the subgroup S4 generated by H and the homographies

n1 : z �→ z + 1

z − 1, n2 : z �→ ιz ,

where ι is a square root of −1. We apply the recipe of (2.5). Since n1 ∈ PGL2(K),it acts on H1(K,H) through its action on H, which permutes e1 and e2; thus itmaps (α, β) ∈ (K∗/K∗2)× (K∗/K∗2) to (β, α). The action of n2 on H fixes e2 andexchanges e1 with e1 + e2; to get the action on H1(K,H) we have to multiply bythe class of the cocycle σ �→ n−1

2σn2, that is, σ �→ i

((σ(ι)/ι) e2

). Hence n2 acts on

H1(K,H) by

n2 · (α, β) = (α,−αβ) .

Let Gα,β be the subgroup of K∗/K∗2 generated by −α and −β; it is the image

of H by the homomorphism det : PGL2(K) → K∗/K∗2. If Gα,β∼= (Z/2)2, the

orbit N · (α, β) in (K∗/K∗2) × (K∗/K∗2) has 6 elements, which are the couples(−x,−y) with x, y ∈ Gα,β , x = y. If Gα,β

∼= (Z/2), the orbit has 3 elements, whichare the couples (−x,−y) with x, y ∈ Gα,β , (x, y) = (1, 1). Finally if Gα,β is trivialthe orbit consists only of (−1,−1). Thus the conjugacy classes of subgroups (Z/2)2

in PGL2(K) are parametrized by the subgroups G ⊂ K∗/K∗2 of order ≤ 4, withthe property (−α,−β)2 = 0 for each α, β in G.

Case 4): The group Dr is generated by two elements s, t with the relationss2 = tr = 1 and sts = t−1. We choose a primitive r-th root of unity ζ and considerthe embedding i : Dr ↪→ PGL2(K) such that i(s) is the involution z �→ 1/z and i(t)the homothety z �→ ζz. The centralizer is Z/2, generated by the involution z �→ −z.As in case 2) it follows that Embi(Dr,PGL2(K)) is isomorphic to H1(K,Z/2). Alsothe previous argument shows that the embedding corresponding to α ∈ K∗ is theconjugate of i by the homography z �→ α′z, with α′2 = α, so it maps s to z �→ α/zand t to z �→ ζz.

To complete the picture we have to take into account the action of the normal-izer N of i(Dr) in PGL2(Ks). This is the subgroup D2r generated by i(s) : z �→ 1/zand the homothety n : z �→ ηz, where η ∈ Ks is a primitive 2r-th root of unity. Theaction of i(s) is trivial, and n acts by multiplication by the cocycle σ �→ n−1 σn,which corresponds to the class of η2 in K∗/K∗2. Since η2 generates μr(K), theassertion 4) follows. �

References

[B] A. Beauville: p-elementary subgroups of the Cremona group, J. of Algebra 314 (2007), 553–564.

[D] J. Dieudonne: La geometrie des groupes classiques. Springer-Verlag, Berlin-Gottingen-Heidelberg, 1955.

[I] I. Isaacs: Character theory of finite groups. AMS Chelsea Publishing, Providence, RI, 2006.

28


[S1] J.-P. Serre: Corps locaux. Hermann, Paris, 1962.[S2] J.-P. Serre: Proprietes galoisiennes des points d’ordre fini des courbes elliptiques, Invent.

Math. 15 (1972), no. 4, 259–331.[S3] J.-P. Serre: Galois cohomology. Springer-Verlag, Berlin, 1997.

Laboratoire J.-A. Dieudonne UMR 6621 du CNRS, Universite de Nice, Parc Valrose,

F-06108 Nice cedex 2, France


29


Picard Groups of Moduli Spaces of Torsionfree Sheaves onCurves

Usha N. Bhosle

Abstract. This article is a short survey of results on the Picard groups ofthe moduli spaces of semistable torsionfree sheaves on irreducible projectivecomplex curves with at most ordinary nodes as singularities.

1. Introduction

In this article we present some old and new results on the Picard groups ofthe moduli spaces of torsionfree sheaves on curves. X and Y will always denotereduced, irreducible, projective, complex curves. A singular curve will be oftendenoted by Y and its normalisation by X. Let U(n, d) be the projective modulispace of S-equivalence classes of semistable torsionfree sheaves of rank n, degreed on X and U ′(n, d) its open subset corresponding to locally free sheaves (vectorbundles). The superscript s will denote the open subsets corresponding to stablepoints. After a short discussion of the case n = 1, we deal with the case n ≥ 2 inthe rest of the paper. The first computations of Picard groups of U(n, d), n ≥ 2,were carried out by C.S.Seshadri for n and d coprime and X a smooth curve. Wepresent here a proof by Ramanan [R] expanding on the ideas of Seshadri. Theseresults were extended to the noncoprime case by Drezet and Narasimhan [DN].Generalizations to singular curves were carried out by us in a series of papers [B3],[B4], [B5]. We state the main results and briefly sketch their proofs. For details,the reader may refer to the original proofs.

2. The rank 1 case

The moduli spaces U ′(1, d) and U(1, d) are respectively the generalized Jacobianand the compactified Jacobian of degree d of the curve X and are denoted by Jd

X

and Jd

X respectively. JX = J0X is the component Pic0(X) of the Picard group Pic X

of X containing the identity.

2010 Mathematics Subject Classification. 14H60, 14D20, 14F05.Key words and phrases. Vector bundles, torsionfree sheaves, moduli spaces, Picard groups.I would like to thank the referee and P.E. Newstead for a very careful reading of the first

version and useful suggestions.


1



31

2 USHA N. BHOSLE

When X is smooth, U(1, 0) = U ′(1, 0) is the Jacobian JX of X, it is an Abelianvariety. Fixing a line bundle L of degree 1 on X, there is the Abel map

AL : X → JX , defined by x �→ L(−x).

The induced pull-back map on line bundles gives an isomorphism

A∗L : Pic0(JX) ∼= JX

independent of the choice of L. Thus JX is a selfdual Abelian variety (see [L,Theorem 3, p.156]; [M, Proposition 6.9, p.118]).

For a singular curve X, JX is not an Abelian variety, nor is the projectivevariety JX , the compactified Jacobian of X. The Abel map is defined as the map

AL : X → JX , defined by x �→ Ix ⊗ L,

where Ix is the ideal sheaf of x ∈ X. Esteves, Gagne and Kleiman have the followinggeneralization of the above result on the selfduality of the Jacobian of a smoothcurve [EGK].

Theorem 2.1. (1) Suppose that a singular curve Y has surficial sin-gularities, that is, singularities of embedding dimension 2. Then A∗

L :

Pic0(JY ) → JY has a right inverse independent of the choice of L i.e.there is f : JY → Pic (JY ) such that A∗

L ◦ f = 1JY. A∗

L is itself indepen-dent of the choice of L.

(2) Let Y be a curve with double points as singularities, i.e. with nodes orcusps. Then A∗

L is an isomorphism.

In fact a relative version of the theorem is proved for flat, projective familiesof geometrically integral curves over an arbitrary locally Noetherian base scheme[EGK].

The compactified Jacobian of a nodal curve is a seminormal variety. Recallthat a variety is called seminormal if all its local rings are seminormal. If B is thenormalisation of a ring A, the ring A is said to be seminormal if it contains eachb ∈ B such that b2, b3 ∈ A. The following general result is very useful for computingPicard groups of seminormal varieties.

Proposition 2.2. [B5, Proposition 2.5] Let U be a seminormal variety (notnecessarily projective) and W the non-normal locus of U , i.e. the scheme of non-

normal points of U . Let π : U → U be the normalisation of U and W the inverseimage of W in U . Define f1 : Pic U → Pic U ⊕ Pic W by f1(L) = (π∗L,L|W ).

Define f2 : Pic U ⊕ Pic W → Pic W by f2(L1, L2) = L1|W ⊗ (π∗L2)∗, for

L1 ∈ Pic U , L2 ∈ Pic W . Then:(1) There is an exact sequence

Pic Uf1→ Pic U ⊕ Pic W

f2→ Pic W .

(2) Assume further that U is projective, U,W are connected and W has p connectedcomponents. Then one has an exact sequence

0 → ×p−1Gm → Pic Uf1→ Pic U ⊕ Pic W

f2→ Pic W .

We now apply Proposition 2.2 to compute the Picard groups of JY , the gener-alized Jacobian, and JY , the compactified Jacobian, of a nodal curve Y with onenode y. JY is a seminormal projective variety containing the smooth subvariety JY .

32

PICARD GROUPS OF MODULI SPACES OF TORSIONFREE SHEAVES ON CURVES 3

It has a natural desingularisation h : JY → JY . The map h induces an isomorphismof JY with h−1(JY ), we identify JY with h−1(JY ) ⊂ JY .

Let X be the normalisation of Y and x, z the points of X lying over y. Thevariety JY ∼= P(Px ⊕ Pz) is a P1-bundle over JX [AK], Px,Pz being line bundles

on JX . Let p′ : JY → JX be the canonical map. JY − JY = D1 ∪D2, where Di aredivisors such that p′ |Di

: Di → JX are isomorphisms. h maps each of D1 and D2

isomorphically onto the smooth Weil divisor JY − JY . One has

Pic JY ∼= p′∗(Pic JX)⊕ Z OP(Px⊕Pz)(1)

∼= Pic JX ⊕ Z.

We shall write the operation in Pic additively.

Lemma 2.3. [B5, Lemma 3.1] One has

Pic JY ∼= ( Pic JX)/(Pz − Px),

where (Pz − Px) denotes the subgroup of Pic JX generated by Pz − Px.

Proof. Since JY is smooth and JY is open in JY the restriction map Pic JY →Pic JY is surjective. Hence we have an exact sequence

1 → H → Pic JY → Pic JY → 0.

Since JY − JY is the union of two irreducible disjoint divisors D1, D2 the kernelH of the restriction map is generated by O(D1) = O(1)− Px,O(D2) = O(1)− Pz

in Pic JX ⊕ Z ∼= Pic JY . Let pZ denote the projection to the Z-factor. ThenpZ(aO(D1) + bO(D2)) = 0 if and only if a + b = 0. Hence Ker (pZ |H) is thesubgroup generated by (Px − Pz) in Pic JX . Also pZ |H is surjective (with eachgenerator O(Di) mapping to 1 ∈ Z). It follows that

H ∼= (Px − Pz)⊕ Z

and

Pic JY ∼= Pic JX/(Px − Pz).

This completes the proof of the lemma. �

For u ∈ X let tu denote translation by OX(u) in JX . Denote by HJ thesubgroup of Pic JX ⊕ Z defined by

HJ = {(L,m) | L ∈ Pic JX ,m ∈ Z, t∗z−xL− L = m(Px − Pz)}.

Proposition 2.4. [B5, Proposition 3.2] There is an exact sequence

1 → Gm → Pic JY → HJ → 0.

Proof. We sketch the idea of the proof. JY is a seminormal variety withJY − JY its non-normal locus. By Proposition 2.2, we have an exact sequence

1 → Gm → Pic JYf1→ Pic JY ⊕ Pic (JY − JY )

f2→ Pic D1 ⊕ Pic D2.

We have

Pic JY = p′∗(Pic JX)⊕ ZO(1),

Pic D1 = (p′|D1)∗Pic JX , Pic D2 = (p′|D2

)∗ Pic JX .

The map p′∗ is an injection and (p′|D1)∗, (p′|D2

)∗ are both isomorphisms. We showthat Ker f2 is isomorphic to the subgroup HJ of Pic JX ⊕ Z. �

33

4 USHA N. BHOSLE

Remark 2.5. The connected component Pic0JY of Pic JY containing theidentity is isomorphic to JY (Theorem 2.1; [EGK]). Under these identifications,the restriction to Pic0JY of the sequence in the statement of Proposition 2.4 is theexact sequence

1 → Gm → JY → JX → 0,

where the map JY → JX is the pull-back π∗ under the normalisation map π.

3. Moduli of vector bundles of rank ≥ 2

Let X be a nonsingular, projective, complex curve of genus g ≥ 2, n, d integers,n ≥ 2. Let UL(n, d) ⊂ UX(n, d) be the subvariety consisting of vector bundleswith fixed determinant L. The first result on the Picard groups of moduli spacesof vector bundles of rank ≥ 2 is due to Seshadri. We sketch here a proof of thisresult given by Ramanan [R] following Seshadri’s ideas.

Theorem 3.1. Let (n, d) = 1. Then

Pic UL(n, d) ∼= Z.

Proof. (Proof sketch following the proof of [R, Proposition 3.4].)Take F ∈ UL(n, d) and E ∈ UL′(n− 1, d′), d′ an integer. Choose a line bundle

M of sufficiently high degree such that there is an injective homomorphism i : E →F ⊗M , M satisfying certain properties [R, Lemma 3.1]. Note that F ⊗M/i(E) ∼=Mn ⊗ L⊗ L′−1. Then

Y := PH1(X,M−n ⊗ L′ ⊗ L−1 ⊗ E)

parametrizes a family of vector bundles of rank n, determinant L and F occursin this family. Let Y s ⊂ Y be the subset corresponding to stable bundles inthis family, then F ∈ Y s. By the universal property of moduli spaces, there is amorphism λ : Y s → UL(n, d). Since Pic Y → Pic Y s is surjective, rank Pic Y s ≤ 1.Let F be the universal bundle on UL(n, d)×X. Define

V = pUL∗(Hom(p∗X(E),F ⊗ p∗X(M))

and let Z ⊂ P(V ) be the subset consisting of injective homomorphisms. Then thereis an isomorphism Y s → Z and λ factors through Z. Hence

Pic UL(n, d) → Pic Z ∼= Pic Y s.

One shows that P(V )− Z is irreducible [R, Lemma 3.5], so that

rank Pic Z + 1 ≥ rank Pic P(V ) = rank Pic UL(n, d) + 1,

i.e. rank Pic UL(n, d) ≤ rank Pic Z = rank Pic Y s ≤ 1.

Since UL(n, d) is projective, rank Pic UL(n, d) ≥ 1. Hence

rank Pic UL(n, d) = rank Pic Z = rank Pic Y s = 1.

Then it follows that Pic UL(n, d) ∼= Pic Z ∼= Z. �

This result was extended to the noncoprime case by Drezet and Narasimhan[DN].

Theorem 3.2. [DN, Theorems A,B and C].Let X be a smooth, projective curve.

(1) Pic UL(n, d) ∼= Z.

34


(2) Pic UX(n, d) ∼= Pic JdX ⊕ZLX , where LX is the determinant line bundle

on UX(r, d).(3) UX(n, d) and UL(n, d) are locally factorial.

We shall give the proof of Theorem 3.2 as a part of the proof of a more gen-eral result on nodal curves. More generally, we have the following result on anyirreducible reduced projective curve.

Theorem 3.3. [B3, Proposition 2.3]. Let Y be any irreducible projective curve

of arithmetic genus gY ≥ 2. Let n ≥ 2 be an integer. Let U′sL (n, d) denote the

moduli space of stable vector bundles on Y of rank n and determinant isomorphicto a fixed line bundle L. Then

Pic U′sL (n, d) ∼= Z or Z/mZ.

Proof. The proof is on similar lines to that of [DN] in the case of a nonsingularcurve, one needs some modifications in the singular case. We briefly sketch theproof.

By tensoring by a line bundle, we may assume that d >> 0. If E is a semistablevector bundle of high degree and determinant L, then E is globally generated andis given by an exact sequence

0 → On−1Y → E → L → 0.

Let P = P(H1(L∗⊗Cn−1)). There is a family E of vector bundles of rank n, degreed on P × Y . Let Ps = {p ∈ P | Ep stable}. By the universal property of moduli

spaces, there is a canonical surjective morphism fE,L : Ps → U′sL (n, d). One shows

that the induced map Pic U′sL (n, d) → Pic Ps (∼= Z or Z/mZ) is an injection by

using an alternative construction of Ps and the construction of the moduli space asfollows.

The moduli space U′sY (n, d) is a GIT quotient of an irreducible nonsingular open

subset R′s of a suitable quot scheme by PGL(q) (see [N1], Remark, p.167, Chapter

5, §7). Let F0 be the universal quotient bundle on R′s × Y, F0 := pR′

s∗(F0) is avector bundle. Let Gr0 be the Grassmannian of subspaces V of dimension n − 1of F0. There exists a geometric quotient Γ0 = Gr0/PGL(q) ([DN], 7.3.1). One

has Pic Gr0 = Pic R′s ⊕ ZOGr0(1) and hence Pic Γ0 = Pic U

′sY (n, d) ⊕ Z.OΓ0

(a),where a = gcd(n, d) and OΓ0

(a) is a line bundle on Γ0 ([DN], Proposition 7.2). LetGr′0 ⊂ Gr0 be the open subset corresponding to vector bundle injections OX⊗V ↪→F0,r, r ∈ R′

s, V ⊂ H0(F)0,r and Gr′0/PGL(q) = Γ′0. Then Γ0 − Γ′

0 is an irreduciblehypersurface [DN, Corollary 7.4]. Computing its ideal sheaf one shows that

0 → Pic U′sY (n, d) → Pic Γ′

0 → Z/(d/a)Z → 0

is exact. Let UGr0 be the universal (relative) subbundle on Gr0. Let

T ′ = Isom(Gr0 ⊗ Cn−1, UGr0), T

′ = T ′/GL(q) = P(T ′)/PGL(q).

T′ is an open subset of T where

T = P(Hom(Gr0 ⊗ Cn−1, UGr0))/PGL(q),

a locally trivial projective bundle over Γ0. Since the ideal sheaf of T − T′ in T isOT(n− 1)⊗ p∗Γ′

0M , where M is a line bundle on Γ0, one has

0 → Pic Γ′0 → Pic T

′ → Z/(n− 1)Z → 0

35

6 USHA N. BHOSLE

exact ([DN], 7.8). Let T′L be the inverse image (in T′) of U ′

L(n, d) ⊂ U ′Y (n, d).

Then there is an exact sequence

0 → Pic U′sL (n, d) → Pic T

′L → Z/(n− 1)

d

aZ → 0.

The final observation is that T′L∼= Ps

L ([DN], Proposition 7.9).�

3.1. Notation. Henceforth Y denotes an irreducible reduced curve with or-dinary nodes as its only singularities (unless otherwise stated) and p : X → Yits normalisation. Let g be the genus of the normalisation X and gY the arith-metic genus of Y . Let UY (n, d) (resp. U

′Y (n, d)) be the moduli space of torsionfree

semistable sheaves (resp. semistable vector bundles) of rank n and degree d on Y .Let U ′

L(n, d) be the closed subset of U ′Y (n, d) corresponding to vector bundles with

fixed determinant L. U ′L(n, d), U

′Y (n, d) are normal quasiprojective varieties. Let

U′sY (n, d) ⊂ U ′

Y (n, d) and U′sL (n, d) ⊂ U ′

L(n, d) be the open subsets correspondingto stable vector bundles. For a line bundle L on Y , let UL(n, d) be the closure ofU ′L(n, d) in UY (n, d).

Theorem 3.4. [B3, Theorem I] Assume that g ≥ 2 and if g = n = 2 then d isodd. Then one has the following.

(1) Pic U′sL (n, d) ∼= Z.

(2) Pic U ′L(n, d)

∼= Z .

Proof. In view of Theorem 3.3, to prove Part (1) we need only to show that

Pic U′sL (n, d) has rank ≥ 1.

Suppose that Y is nonsingular. Then U ′L(n, d) is projective, hence its Pi-

card group has rank at least 1. Under the assumptions of the theorem, one hascodimU ′

L(n,d) (U′L(n, d)−U

′sL (n, d)) ≥ 2. Since U ′

L(n, d) is normal, this implies that

Pic U′

L(n, d) ⊂ Pic U′sL (n, d) and hence the theorem follows from Theorem 3.3. This

is essentially the proof of [DN] in the nonsingular case.If Y is nodal, U ′

L(n, d) is not projective, so one has to work harder. Moreover, itis impossible to do the above codimension computations on Y (since tensor productsof semistable torsionfree sheaves on Y are not semistable, etc.). Hence we use thenormalisation ML(n, d) of UL(n, d) [B1]. ML(n, d) is the moduli space of general-ized parabolic bundles of rank n, degree d and determinant isomorphic to L = p∗(L)on the normalisation X of Y . There is a rational map ML(n, d) → UX,L(n, d), it

induces an isomorphism M′sL (n, d) ∼= U

′sL (n, d). Under the assumptions of the the-

orem, it is proved that the generator of Pic UX,L(n, d) gives a nontorsion element

in Pic M′sL (n, d) and hence in Pic U

′sL (n, d)). Thus Pic U

′sL (n, d) has rank ≥ 1 [B3,

Proposition 2.2]. This proves Part (1).Now if we can show that

(3.1) codimU ′L(n,d) (U

′L(n, d)− U

′sL (n, d)) ≥ 2,

the normality of U ′L(n, d) will imply that Pic U

′

L(n, d) ⊂ Pic U′sL (n, d) and hence

Pic U′

L(n, d) has rank 1, proving Part 2 (in fact the restriction map Pic U′

L(n, d) →Pic U

′sL (n, d) can be shown to be an isomorphism [B3, Proposition 3.6]). Using

ML(n, d) and the fact that codimUX,L(n,d) (UX,L(n, d)− UsX,L(n, d)) ≥ 2 for g ≥ 2

(except for g = n = 2, d even), one can show that the inequality (3.1) holds underthese conditions ([B3], Corollary 1.7).

36

�


Theorem 3.5. [B4, Theorem 3A] With the same assumptions as in Theorem3.4, let Jd

Y be the generalized Jacobian of Y of degree d i.e. the moduli variety ofdegree d line bundles on Y . Then(a) Pic U

′sY (n, d) ∼= Pic Jd

Y ⊕ Z,(b) Pic U ′

Y (n, d)∼= Pic Jd

Y ⊕ Z,(c) U ′

Y (n, d) and U ′L(n, d) are locally factorial.

Proof. (a) and (b) are proved using the fact that the determinant map inducesan injection on Picard groups and the fibres of the determinant map are isomorphicto U

′

L(n, d).

(c) Since U′sL (n, d) is smooth, its Picard group and class group coincide. The

normality of U′

L(n, d) and the inequality (3.1) imply that the class groups of U′

L(n, d)

and U′sL (n, d) are isomorphic. Moreover, Pic U

′

L(n, d) → Pic U′sL (n, d) is injective.

Hence it is easy to see that to show that U′

L(n, d) is locally factorial, it suffices to

show that the restriction map Pic U′

L(n, d) → Pic U′sL (n, d) is surjective. This is

proved using the fact that the determinant line bundle generates Pic U′sL (n, d) and

comes from Pic U′

L(n, d) ([B3], p. 262).It is easy to prove the local factoriality of U ′

Y (n, d) by similar arguments using(a) and (b). �

4. Moduli of torsionfree sheaves on nodal curves.

We start with the simple case of curves with lower genera. For low genera andranks, there are explicit descriptions of moduli spaces of torsionfree sheaves. Thismakes the computation of their Picard groups much easier.

4.1. Case gY ≤ 2. .Any irreducible nonsingular (respectively nodal) curve Y with gY = 1 is a

nonsingular (respectively nodal) Weierstrass curve. If Y is nonsingular, it is anelliptic curve. If Y is nodal, it has normalisation P1 and a unique singular pointwhich is an ordinary node. If gcd(n, d) = h, the moduli space U(n, d) is isomorphicto Sh(Y ), the hth symmetric product of Y (see [At], [Tu] for the nonsingular case;[BBH, Corollary 6.47], [HLST, Remark 1.27] for the singular case). In particular,for (n, d) = 1, UY (n, d) is isomorphic to Y . For Y nodal, U ′

Y (n, d) is isomorphic tothe variety of nonsingular points of Y i.e. to the affine variety A

1 − 0.

Proposition 4.1. Let gY = 1.(i) If (n, d) = 1, then Pic UY (n, d) ∼= Pic Y .If Y is singular, then Pic UY (n, d) ∼= Gm ⊕ Z and Pic U ′

Y (n, d), Pic UL(n, d) aretrivial.(ii) If h = (n, d) > 1 then Pic UY (n, d) ∼= Pic Sh(Y ).For L ∈ JY , Pic UL(2, 0) ∼= Z.If Y is singular, then Pic U ′

L(2, 0) and Pic U ′Y (2, 0) are trivial.

Proof. By the explicit description, Pic UY (n, d) ∼= Pic Sh(Y ). If (n, d) = 1,then UL(n, d) is a point, so its Picard group is trivial. If Y is nodal, then

Pic Y ∼= Gm ⊕ Pic P1 ∼= Gm ⊕ Z

37

8 USHA N. BHOSLE

and since U ′Y (n, d)

∼= A1 − 0, it has a trivial Picard group.For the results in rank 2 note that for L = O, every E ∈ UL(2, 0) is S-equivalent

to N ⊕N∗ for N ∈ JY . Hence UL(2, 0) is isomorphic to the quotient of Y by Z/2Zwhich is isomorphic to P1. Moreover, U ′

L(2, 0)∼= A1 and U ′

Y (2, 0) is fibred overJY ∼= A

1 − 0 with fibres U ′L(2, 0). For more details, see [B4, Proposition 2.3]. �

For gY = 2, Y is a hyperelliptic curve. If Y is singular, then Y has one or twonodes, the normalisation is an elliptic curve or a projective line. Suppose that d isodd. If Y is nonsingular, then UL(2, d) is isomorphic to a nonsingular intersectionQ of two quadrics ([N], [NRa, Theorem 4]). If Y is nodal, then the study of theextension of the determinant map U ′(2, d) → Jd

Y to UY (2, d) [B1] shows that thesingular set UL(2, d)−U ′

L(2, d) consists of direct images of stable vector bundles ofrank 2 and a suitable fixed determinant on the (partial) normalisations of Y . Notethat the (partial) normalisations are curves of arithmetic genus 1 and the projectiveline. By Grothendieck’s theorem, there are no stable vector bundles on a projectiveline. On a curve of arithmetic genus 1, there is a unique stable vector bundle ofrank 2 and a fixed determinant of odd degree. It follows that UL(2, d) − U ′

L(2, d)consists of one or two singular points according as Y has one or two nodes.

If d is even, one has U ′L(2, d) = UL(2, d) ∼= P

3 and the subset UL(2, d)−UsL(2, d)

is isomorphic to the Kummer variety in P3 corresponding to Q considered as aquadratic complex of lines in P3 ([NRa, Theorem 2], [B2, Corollary 3.5]).

Lemma 4.2. The Kummer variety K is an irreducible quartic surface in P3.

Proof. The curve Y is associated to a pencil of quadrics in k6 with the fol-lowing standard equation (k = C in our case, in general k can be an algebraicallyclosed field of characteristic = 2).

q1 =I∑

i=1

2XiYi +∑

j≥I+1

X2j

q2 =

I∑i=1

(λiXiYi +X2i ) +

∑j≥I+1

λjX2j

Here I = 0, 1 or 2 according as Y is nonsingular, has one node or two nodes.The intersection of quadrics can be considered as a quadratic complex of lines

in P3 of type [111111], [21111], [2211] in the three cases. For the first type, thesingular surface K is the classical Kummer surface, which is an irreducible quarticsurface with 16 nodes. In the other two cases, the equation of K may be found, forinstance, in Article 271 on p.211 and article 186 on p.219 of [J]; in both cases K isa quartic which is easily checked to be irreducible.

�Proposition 4.3. Let gY = 2. Then

(i) Pic U ′L(2, d) = Pic UL(2, d) ∼= Z for d even.

(ii) Pic U′sL (2, d) ∼= Z/4Z for d even.

(iii) Pic U ′L(2, d) = Pic U

′sL (2, d) ∼= Z for d odd.

(iv) Pic U ′Y (2, d)

∼= Pic JY ⊕ Z for d odd.(v) Pic UY (2, d) ∼= Pic JY ⊕ Z for d even and Y nonsingular.

Proof. (i) follows immediately from the fact U ′L(2, d) = UL(2, d) ∼= P3.

(ii) Since U ′L(2, d)

∼= P3 is nonsingular there is a surjective homomorphism of Picard

38


groups Pic U ′L(2, d) → Pic U

′sL (2, d). As the complement of U

′sL (2, d) in U ′

L(2, d)is a divisor of the line bundle OP3(4) (by Lemma 4.2), the kernel of the surjective

homomorphism is 4Z. Hence Pic U′sL (2, d) ∼= Z/4Z for d even.

(iii) In case Y is nonsingular, Q is a nonsingular complete intersection in P5 ofdimension 3. By [H, Corollary 3.2, p.179], Pic Q ∼= ZOQ(1), proving the result.Alternatively, the result also follows from Theorem 3.3 as Us

L(2, d) = UL(2, d) is aprojective variety.

Suppose that Y is nodal. Let p : UL(2, d) → UL(2, d) be a normalisation. SinceU ′L(2, d) is normal (in fact nonsingular), p is an isomorphism over U ′

L(2, d). As thesingular set UL(2, d)− U ′

L(2, d) consists of one or two points, it follows that

codim UL(2, d)− p−1U ′L(2, d) = codim UL(2, d)− U ′

L(2, d) ≥ 3.

Since UL(2, d) is normal, this implies that

Pic UL(2, d) ↪→ Pic p−1U ′L(2, d)

∼= Pic U ′L(2, d).

Since UL(2, d) is projective, so is UL(2, d) and hence

rank Pic U ′L(2, d) ≥ rank Pic UL(2, d) ≥ 1 .

Now it follows from Theorem 3.3 that Pic U ′L(2, d)

∼= Z.(iv) This follows from (iii) as in the following Part(v) (or as in [B4, Theorem 3A]).(v) The morphism det : UY (2, d) = U ′

Y (2, d) → JdY is a P3 fibration (not locally

trivial). Hence for any line bundle M on JdY , one has det∗(det

∗M) = M , so thatdet∗ is injective on line bundles. Let N be a line bundle on UY (2, d). The restrictionof N to the fibre over L is O(mL),mL ∈ Z. Jd

Y being irreducible, mL = m must beconstant. The allowable values of m form a non-trivial subgroup of Z, so restrictionto a fibre determines a surjection Pic UY (2, d) → Z. By the seesaw theorem, thekernel of this surjection is isomorphic to Pic Jd

Y . This completes the proof. �

4.2. Case gY ≥ 3. .In the rest of this section, we assume that Y is a nodal curve of genus gY ≥ 3

with one node y and x, z points of the normalisation X lying over y. Our aim isto compute the Picard groups of UY (2, d), UL(2, d) and U ′

L(2, d). For gY ≥ 3, thevariety UY (2, d) is not normal, it is seminormal [NR].

Theorem 4.4. [B4, Proposition 2.8] For gY ≥ 3, one has:

(1) U′sL (2, d) ∼= Z.

(2) U ′L(2, d)

∼= Z.

Proof. We write U′sL = U

′sL (2, d), U ′

L = U ′L(2, d). Let p : UL → UL be a

normalisation, it is an isomorphism over U ′L (since U ′

L is normal). Then codim

UL − p−1U ′L = codim UL − U ′

L ≥ 3 [B4, Lemma 2.5]. Since UL is normal, this

implies that Pic UL ↪→ Pic (p−1U ′L)

∼= Pic U ′L. Since UL is projective, so is

UL and hence rank (Pic UL) ≥ 1, rank (Pic U ′L) ≥ 1. Since U ′

L is normal and

codim (U ′L − U

′sL ) ≥ 3 [B4, Lemma 2.4], we have Pic U ′

L ↪→ Pic U′sL . Thus rank

(Pic U′sL ) ≥ 1. Now it follows from Theorem 3.3 that Pic U

′sL

∼= Z and hencePic U ′

L∼= Z. �

The variety UY (2, d) has a filtration

U := UY (2, d) = W2 ⊃ W1 ⊃ W0

39

10 USHA N. BHOSLE

where Wi are seminormal, the non-normal locus of Wi is Wi−1 for i = 1, 2 andW0

∼= UX(2, d− 2) is smooth. Let U ′ = W2 −W1, U1 = W1 −W0, U0 = W0 be thestrata (for more details, see [NR] or [B5, section 4.2]). Denote by U1,L(2, d) thesubvariety of U1(2, d) corresponding to torsionfree sheaves with a fixed (not locallyfree) determinant [B4, 2.10].

Theorem 4.5. [B4, Theorem 2 and Theorem 3B] Let gY ≥ 2 and let L be atorsionfree sheaf of rank 1 on Y which is not locally free. Then(a) Pic U1,L(2, d) ∼= Z.Moreover, assume that d is odd if gY = 2. Then(b) Pic Us

1,L(2, d)∼= Z, Pic Us

1 (2, d)∼= Pic JX ⊕ Z.

(c) Pic U1(2, d) ∼= Pic JX ⊕ Z.(d) U1(2, d) is locally factorial.

Proof. This is proved as in the case of U ′L(2, d), U

′(2, d). �

To compute the Picard group of U using our Proposition 2.2, we need to knowthe Picard group of the normalisation P of U and that of the non-normal locusW1. The normalisation P was constructed by Narasimhan and Ramadas [NR]as the moduli space of GPS (generalized parabolic sheaves) on the normalisationX of Y . A GPS is a sheaf E, which is torsionfree outside {x, z} ⊂ X, togetherwith a 2-dimensional quotient Q of Ex ⊕ Ez. A GPS is called a GPB (generalizedparabolic bundle) if E is a vector bundle. The moduli space P is a normal projective(irreducible) variety with rational singularities. It has a filtration

P = P 2 ⊃ P 1 ⊃ P 0.

We have P 1 = D1 ∪ D2 where D1,D2 are irreducible and normal Cartier divisors[NR]. P 0 = (D1 ∩ D2)

∐D1(0)

∐D2(0) where Di(0) are closed subsets of Di, i =

1, 2. Let

P ′ = P − P 1, P1 = P 1 − P 0, P0 = P 0.

The normalisation map p : P → U maps P i onto Wi. Its restriction to P ′ is anisomorphism onto U ′. Moreover, for each j, the restriction

p |Dj: Dj → W1

is a normalisation map with D′j = Dj − (D1 ∩D2)−Dj(0) mapping isomorphically

onto U1 = W1 −W0.

Theorem 4.6. [B5, Theorem I] For gX ≥ 2, rank 2 and odd degree d, one hasthe following.

(1) P − P0 is nonsingular.(2) Pic (P − P0) ∼= Pic UX(2, d)⊕ Z.(3) Pic P ∼= Pic Jd

X ⊕ Zp∗LY ,LY being the determinant line bundle on U and p : P → U the normali-sation map. The divisor class group of P (and of P − P0) is isomorphicto Pic UX(2, d)⊕ Z.

Idea of Proof. Part(1) is proved by using the smooth determinant map [B1]

detp : P − P0 → JY

and proving that all its fibres are smooth [B5, Proposition 4.6].

40


For (2), let E → UX(2, d)×X be the universal bundle. Define Ex = E|UX (2,d)×x,Ez = E|UX(2,d)×z and let

pr : (G := Gr(Ex ⊕ Ez)) → UX(2, d)

be the Grassmannian bundle of two dimensional quotients of Ex ⊕ Ez. Then Gparametrizes a family of GPBs giving a morphism Gs → P . One shows that itinduces an isomorphism from a big open subset of G onto a big open subset ofP −P0, here big means the complement has codimension ≥ 2 [B5, Proposition 4.6].

The proof of (3) is far more complicated [B5, Proposition 4.14]. Since P isnormal and P0 is of codimension ≥ 2, Pic P ↪→ Pic(P − P0), but computation ofthe image is hard. We show that

Pic Di∼= Pic Jd

X ⊕ Z ∼= Pic Di(0)

and study the restriction maps from Pic P to these schemes. �

Using the fact that W1 is seminormal with normalisation D1 and non-normallocus W0, by applying Proposition 2.2, we compute the Picard group of W1 (forodd degree d). We partially succeed in computing the Picard group of U similarly.

Theorem 4.7. [B5, Theorem II].

(1) Pic W0∼= Pic JX ⊕ Z.

(2) Pic W1 is given by an exact sequence

1 → Gm → Pic W1 → H ′W1

⊕ Z → 0

where H ′W1

is isomorphic to the subgroup G of Pic JdX defined by

G = {L ∈ Pic JdX | (⊗OX(z − x))∗L = L}.

(3) One has an exact sequence

1 → Gm → Pic UresW1→ Pic W1.

Proof. (1) is proved using the isomorphism W0∼= UX(2, d−2) [B5, Corollary

4.11].(2) is proved in [B5, Proposition 4.15] using (1) and Proposition 2.2.(3) is deduced from Proposition 2.2 using (2) and a computation of Pic D1 [B5,Proposition 4.17]. �

Remark 4.8. Computing the image HU of Pic U in Pic W1 explicitly seemsdifficult as we do not have enough information about Pic P1. One has

HU = {B ∈ Pic W1 | p∗(B) = (Det∗L+m LY ) |P1, L ∈ G,m ∈ Z}.

Here LY = p∗LY , LY being the determinant line bundle on Y . Det denotes thedeterminant morphism P → Jd

X which associates to a GPB (E,Q) the determinantof the underlying vector bundle E. Note that every GPS is S-equivalent to a GPB,so there is a well-defined morphism Det [Su].

41

12 USHA N. BHOSLE

References

[AK] Altman A., Kleiman S. The presentation functor and the compactified Jacobian, TheGrothendieck Festschrift, Vol. I, Progr. Math., 86, Birkhauser Boston, (1990) 15–32.

[At] Atiyah M. F. Vector bundles over an elliptic curve, Proc. London Math. Soc. 7 (1957),412–452.

[BBH] Bartocci, C., Bruzzo, U., Hernandez Ruiperez D. Fourier-Mukai and Nahm transformsin geometry and mathematical physics. Progress in Mathematics 276, Birkhauser (2009).

[B1] Bhosle, Usha N. Generalised parabolic bundles and applications to torsion free sheaveson nodal curves, Arkiv for Matematik 30 (1992), No.2, 187–215.

[B2] Bhosle Usha N. Vector bundles of rank two and degree zero on a nodal curve, Proc.Barcelona-Catania Conferences. 1994–1995. Lecture notes in pure and applied Mathe-matics, Vol.200, Marcel-Dekker (June 1998).

[B3] Bhosle Usha N. Picard groups of the moduli spaces of vector bundles, Math. Ann. 314(1999), 245–263.

[B4] Bhosle Usha N. Picard groups of the moduli spaces of semistable sheaves, Proc. IndianAcad. Sci. (Math. Sci.) 114 (2004), no. 2, 107–122.

[B5] Bhosle Usha N. Seminormal varieties, torsionfree sheaves, and Picard groups, Commu-nications in Algebra, 36 (2008), no. 3, 821–841.

[DN] Drezet J.-M., Narasimhan M.S. Groupe de Picard des varietes de modules de fibres semi-stables sur les courbes algebriques, Invent. Math. 97 (1989), 53–94.

[EGK] Esteves E., Gagne M. and Kleiman S. Autoduality of the compactified Jacobian, J. LondonMath. Soc. (2) 65 (2002), no. 3, 591–610.

[H] Hartshorne R. Ample subvarieties of Algebraic varieties, Lecture Notes in Mathematics156, Springer-Verlag (1970).

[HLST] Hernandez Ruiperez D., Lopez Martin A.C., Sanchez Gomez D., Tejero Prieto C. Modulispaces of semistable sheaves on singular genus one curves, arXiv:0806.2034v2.

[J] Jessop C.M. A treatise on the line complex, Cambridge University Press, Cambridge,(1903).

[L] Lang S. Abelian varieties Interscience Tract 7, John Wiley, New York, (1959).[M] Mumford D. Geometric invariant theory, Ergebnisse der Mathematik 34, Springer, Berlin

(1965).[NR] Narasimhan M.S., Ramadas T. Factorisation of generalised theta functions I, Invent.

Math. 114 (1993), 565–623.[NRa] Narasimhan M.S., Ramanan S. Moduli of vector bundles on a compact Riemann surface,

Ann. Math. 89 (1969), no. 1, 19–51.[N] Newstead P.E. Stable bundles of rank 2 and odd degree over a curve of genus 2, Topology

7 (1968), 205–215.[N1] Newstead P.E. Introduction to moduli problems and orbit spaces, Tata Institute of Fun-

damental Research Lectures on Mathematics, Springer-Verlag (1978).[R] Ramanan S. The moduli spaces of vector bundles over an algebraic curve, Math. Ann.

200 (1973), 69–84.[Su] Sun X. Degeneration of moduli spaces and generalized theta functions, J. Algebraic Geom.

9 (2000), No. 3, 459–527.[Tu] Tu L. Semistable vector bundles over elliptic curves, Adv. Math. 98 (1993), 1–26.


Road, Mumbai 400005, India


42


On the Moduli of Orthogonal Bundles on a NodalHyperelliptic Curve

Usha N. Bhosle

Dedicated to Prof. Ramanan on his 70th birthday.

Abstract. Let X be a complete irreducible hyperelliptic curve of arithmeticgenus g with an ordinary node as its only singularity. We find explicit descrip-tions of the moduli spaces of rank 4 orthogonal bundles with a Z/2Z-action(and of a certain fixed topological type) on X in terms of spaces associated tothe singular pencil of quadrics determined by X.

1. Introduction

The moduli spaces UL2 of semistable vector bundles of rank 2 and a fixed deter-

minant L of degree d on a smooth hyperelliptic curve X of genus g have beautifulexplicit descriptions. P.E. Newstead showed that, for g = 2 and d odd, the modulispace UL

2 is isomorphic to a nonsingular intersection Q of two quadrics in P5C([N],

see also [NRa, Theorem 4]). Narasimhan and Ramanan proved that, for g = 2 andd even, UL

2 is isomorphic to P3C= PH0(J, θ2) where θ2 denotes the square of the

theta line bundle on the Jacobian J of X. In my maiden work with Ramanan, wegeneralized these results to smooth hyperelliptic curves of any genus g ≥ 2 definedover an algebraically closed field of characteristic �= 2 [DR]. Generalizations tocharacteristic 2 were carried out by me later ([B3], [B5]). It was soon realizedthat the explicit descriptions for UL

2 in fact follow from the explicit descriptions forthe moduli spaces of rank 4 orthogonal bundles with Z/2Z-action of certain types([R], [B1]). A generalization of the results of [B1] to a hyperelliptic curve withone ordinary node was carried out in [B4]. The aim in this paper is to generalizethe results in [R] to a hyperelliptic curve with one ordinary node.

Let k be an algebraically closed field of characteristic 0. Let X be a completeirreducible reduced hyperelliptic curve of arithmetic genus g ≥ 2 over k with asingle ordinary node as its only singularity. To such a curve we associate a singular

2010 Mathematics Subject Classification. 14H60, 14D20, 14F05.Key words and phrases. Pencils of quadrics, nodal curves, torsionfree sheaves, moduli spaces.The initial part of this work was done during my visits to Liverpool University. I would like

to thank P.E. Newstead for numerous discussions and the Liverpool University for hospitality.


1



43

2 USHA N. BHOSLE

pencil of quadrics with Segre symbol [21 · · · 1] given by

q1 =

2g∑i=1

X2i + 2X0Y0, q2 =

2g∑i=1

aiX2i + (X2

0 + 2a0X0Y0),

with ai distinct scalars. Let R be the scheme of (g − 1) -dimensional subspaces ofk2g+2 which are isotropic for this pencil. Let R0 be the subscheme of R consistingof those subspaces which contain the unique singular point of the intersection ofquadrics of the pencil. Our main result is the following.

Theorem 1.1. The moduli space M of semistable orthogonal bundles of rank4 with a Z/2Z -action (and of a certain fixed topological type) on X is isomorphicto the quotient of R −R0 by (Z/2Z)2g.

In section 2, we study torsionfree sheaves with Z/2Z-actions on X. The pencilof quadrics associated to X is described and some results related to it are given insection 3. The main theorem is proved in section 4.

2. Torsionfree sheaves with Z/2Z-action

Let k be an algebraically closed field of characteristic 0. Let X be an irreduciblecomplete reduced curve over k with a single node as the only singularity. Let gdenote the arithmetic genus of X. It is well known that X is Gorenstein and hencehas a locally free dualising sheaf ωX of rank 1 and degree 2g − 2, where

deg(.) = χ(.)− χ(OX).

Let π : X → X be the normalisation map.

Lemma 2.1. Let E be a torsionfree coherent sheaf on X.

(a) If E is semistable (respectively stable and of rank ≥ 2) and χ(E) > (rankE)(g − 1) (respectively ≥), then H1(E) = 0.

(b) If E is semistable (respectively stable and of rank ≥ 2) and χ(E) < (rankE)(1− g) (respectively ≤), then H0(E) = 0.

Proof. (a) Suppose that E is semistable with χ(E) > (rank E)(g − 1)and H1(E) �= 0 i.e. H0(Hom(E,ωX)) �= 0. Then there exists a nonzerohomomorphism E → ωX . Since E is semistable, χ(ωX) ≥ χ(E)/rank E,i.e. χ(E) ≤ (rank E)(g − 1), a contradiction.

(b) Suppose that E is semistable with χ(E) < (rank E)(1−g). If H0(E) �= 0,then there exists a nonzero homomorphism OX → E. The semistabilityof E implies that χ(E)/ rank E ≥ χ(OX) = 1− g, giving a contradiction.

In the case E is stable, the result follows similarly. �

2.1. Henceforth we assume that X is hyperelliptic, i.e. there exists a finitedegree two map p : X → P1. Let i denote the hyperelliptic involution on X. Theunique node w0 of X must be i-invariant and hence it is a ramification point ofp. Let W0 = {w0, w1, · · · , w2g} be the set of ramification points of X. Let OX(1)denote the line bundle p∗OP1(1).

We define an i-action on a torsionfree coherent sheaf E on X as a Z/2Z-actionon E which lifts the involution i. We call E i-invariant if E ≈ i∗E.

44

ORTHOGONAL BUNDLES ON A NODAL CURVE 3

Lemma 2.2. (a) There is a one-one correspondence between i-invariantline bundles of even degree (respectively odd degree) d and the set of parti-tions of W0 = {w0, w1, · · · , w2g} into two subsets T1, T2 such that the setsT1 ∩ {w1, · · · , w2g} and T2 ∩ {w1, · · · , w2g} have even cardinality (respec-tively odd cardinality).

(b) There is a 1− 1 correspondence between i-invariant torsionfree sheaves ofrank 1 and even degree (respectively odd degree) which are not locally freeand the partitions of W0 − w0 = {w1, · · · , w2g} into two subsets T ′

1, T′2 of

odd (respectively even) cardinality.

Proof. This lemma can be proved as in the case when X is nonsingular [DR],so we omit the details.

(a) The correspondence is given by

L =

⎧⎪⎪⎨⎪⎪⎩

⊗w∈T1

OX(w)⊗OX(−#T1

2 + d2 ) if w0 �∈ T1

⊗w∈T1−w0

OX(w)⊗OX(d+1−#T1

2 )⊗M if w0 ∈ T1,

where M is the unique square root of OX which becomes trivial on pulling

back to the normalisation X. The set T1 is characterised as the set ofpoints in W0 such that i acts on the fibre of L at w by −Id.

(b) Let T (X) be the set of torsionfree sheaves of rank 1 on X which are

not locally free. The map Pic X → T (X) defined by L → π∗L is anisomorphism; moreover L is i-invariant if and only if π∗L is so. Thus

i-invariants in Pic X are in bijective correspondence with i-invariants in

T (X). Using this and the correspondence between i-invariants in Pic Xand partitions of W0 − w0, we get the result. In case degL = 0, one has

L =⊗w∈T ′

1

OX(w)⊗OX(−1−#T ′

1

2 )⊗ π∗(O ˜X),

for T ′1 ⊂ {w1, · · · , w2g},#T ′

1 odd. T ′1 is the set of points w in W0 − w0

such that i acts on the fibre of (π∗L/ torsion) at w by −Id.

�

2.2. The coordinate ring of an affine neighbourhood of w0 in X can be writtenas B = A[y]/(y2 − a), where A is the coordinate ring of an affine neighbourhoodof p(w0) and a ∈ A. The element y is a generator of the maximal ideal at w0 anddefines a Cartier divisor Y such that OY is supported at w0. Since w0 is a node,a ∈ m2,m = maximal ideal at p(w0), and OY ≈ (OP1)/m2. Let I denote the idealsheaf OX(−Y ); it has an i-action given by y → −y.

The i-action on a torsionfree coherent sheaf F induces i-actions on its fibres overthe ramification points, on its cohomology and on the Euler characteristic χ(F ). Let( )− denote the space of anti-invariants for the i-action, i.e. the eigenspace corre-sponding to the eigenvalue (−1) for the i-action. We define rw0

(F ) := dim (F/IF )−

and for w ∈ W0 − w0, rw(F ) := dim (F ⊗ OX/mw)−, where mw is the maximal

ideal at w. Clearly, rw, w �= w0, are additive for exact sequences of torsionfreecoherent sheaves on X. [B4, Lemma 2.2] shows that this is true for rw0

also.

45

4 USHA N. BHOSLE

Proposition 2.3. [B4, Proposition 2.3] Let F be a torsionfree coherent sheafwith an i-action on X. Then

χ(F )− =1

2

[χ(F )− (rank F )(g + 1) +

2g∑i=0

rwi

].

3. The singular pencil

3.1. The singular pencil in P2g+1. Let w0, w1, · · · , w2g be distinct closedpoints in P

1. Consider the divisor

W = 2w0 + w1 + · · ·+ w2g

in P1. We have OW = OP1/m2w0

⊕( 2g⊕

i=1

OP1/mwi

). The multiplication in OP1

induces quadratic maps Ai on OP1/mwiand A0 on OP1/m2

w0given by

Ai(Xi) = X2i , A0(X0, Y0) = (X2

0 , 2X0Y0),

with Xi ∈ OP1/mwi, X0 ∈ OP1/mw0

, Y0 ∈ mw0/m2

w0. We interpret Ai to have

values in (OP1(2g+1))⊗OP1/mwiand A0 to have values in OP1(2g+1)⊗OP1/m2

w0.

The Ai, i ≥ 0 combine to form a quadratic map A with values in OP1(2g+1)⊗OW .The canonical map Ψ : H0(OP1(2g+1)) → OP1(2g+1)⊗OW can easily be checkedto be an isomorphism. Let e denote the evaluation map

e : P1 ×H0(OP1(2g + 1)) → OP1(2g + 1)

and let ex denote the evaluation map H0(OP1(2g+1)) → (OP1(2g+1))x. We definea quadratic form Q on P

1 × k2g+2 with values in OP1(2g + 1) by Q = e ◦Ψ−1 ◦A,

where A =

2g⊕i=0

Ai. For every x in P1, let Qx = ex ◦Ψ−1 ◦A be the ‘quadratic form

at x’.

Remark 3.1. Notice that for every x, the one-dimensional subspace

D0 = {(0, Y0, 0, · · · , 0) | Y0 ∈ k}

is isotropic for Qx. The quadratic form Qx is nonsingular if x �∈ supp W , i.e. theforms Qwi

, i = 0, · · · , 2g are the only singular members. In fact, Q is dual to thesingular pencil (I) of quadratic forms with Segre symbol [211 · · · 1] given by

(I) q1 =

2g∑i=1

X2i + 2X0Y0,

q2 =

2g∑i=1

aiX2i + (X2

0 + 2a0X0Y0)

with distinct scalars ai.

46


3.2. The variety S. Let S be the variety of (g − 1)-dimensional vector sub-spaces V0 of k2g+2 which are isotropic for the pencil (I). Let D (resp. D0) be thetwo-dimensional (respectively one-dimensional) subspace of k2g+2 with coordinatesX0, Y0 (respectively Y0 only ); thus D0 is the unique subspace of D isotropic forthe pencil and determines the unique singular point of the intersection of quadricsof the pencil (I). In particular, if V0 ∈ S, then V0 ∩ D �= {0} if and only ifV0 ⊃ D0. We have an orthogonal decomposition k2g+2 = D ⊕ C with respectto the pencil (I), {Xi}i>0 being the coordinates in C. The pencil (I) inducesthe dual pencil Q on (k2g+2)∗ ≈ OW . Under the natural correspondence betweenGrassg−1(k

2g+2) and Grassg+3(k2g+2)∗ which maps a subspace V0 to the subspace

V of linear forms on k2g+2 vanishing on V0, the variety S corresponds to the varietyR ⊂ Grassg+3(k

2g+2)∗ consisting of subspaces V such that the orthogonal com-plement of V with respect to a general quadratic form in the pencil is an isotropicsubspace of dimension g − 1. Using the isomorphism k2g+2 → (k2g+2)∗ given by ageneric member of the pencil, say q1, V gets identified with the orthogonal comple-ment of V0 with respect to q1.

Let S0 be the closed subvariety of S consisting of V0 such that V0 ⊃ D0, orequivalently V0 ∩ D �= {0}. Let R0 be the corresponding subvariety of R. Let ⊥denote the orthogonal complement. Now V0 ∩D �= {0} if and only if V ⊥

0 +D⊥ �=k2g+2, i.e. V ⊥

0 + C �= k2g+2, i.e. dim V ⊥0 ∩ C �= g + 1. But

g + 3 ≥ dimV ⊥0 ∩ C ≥ dimV ⊥

0 − 2 = g + 1.

Hence for V0 ⊃ D0, dim V ⊥0 ∩ C = g + 2 or g + 3. Now

dimV ⊥0 ∩ C = g + 3 ⇒ V0 ⊂ V ⊥

0 ⊂ C,

contradicting the assumption V0 ⊃ D0. Thus dim V ⊥0 ∩ C = g + 2 for V0 ⊃ D0.

Hence, if V ∈ R, then V ∈ R0 if and only if dim V ∩ C = g + 2 and V ∈ R−R0 ifand only if dimV ∩ C = g + 1, i.e. V + C = k2g+2. Also for i �= 0, V0 ∩Di = {0}.Hence V ⊥

0 +(⊕

j �=i

Dj

)= k2g+2, i.e.

V +(⊕

j �=i

Dj

)= k2g+2, V + C = k2g+2

for V ∈ R−R0.

Lemma 3.2. The group G of transformations leaving the pencil (I) invariantis isomorphic to (Z/2Z)2g+1.

Proof. For i > 0, let Di denote the subspace of k2g+2 with coordinate Xi.Any transformation which keeps the pencil invariant must keep the spaces D andDi invariant as k

2g+2 = D⊕ (⊕

iDi) is a decomposition orthogonal with respect to

the pencil. Hence G = Π2gi=0Gi where G0 is the group of transformations which keep

the restriction of the pencil to D invariant and for i ≥ 1, Gi denotes the group oftransformations keeping the pencil restricted to Di invariant. Clearly, Gi ≈ Z/2Zfor i > 0. It is easy to check that the only elements of GL(2) which keep both theforms X0Y0 and 2a0X0Y0 +X2

0 invariant are ±Id, i.e. G0 ≈ Z/2Z. �

Remark 3.3. The group G acts on S (or R) with ±Id acting trivially, hencethe group acting on S (or R) is P(G) ≈ (Z/2Z)2g. In view of Lemma 2.2, thereis a 1-1 correspondence between elements of this group and elements of the set

47

6 USHA N. BHOSLE

{i-invariant line bundles of a fixed even degree } ∪ {i-invariant line bundles of afixed odd degree}.

Proposition 3.4. Let M be a vector space, M =⊕

wMw. Let Q1 =∑w

Qw

and Q2 =∑w

Q′w be nondegenerate quadratic forms on M with Qw, Q

′w nonde-

generate quadratic forms on Mw. Let Gk (respectively G′k) ⊂ Grassn(M) be the

subvariety of V ⊂ M such that Q1 (respectively Q2) restricted to V has rank ex-actly k. If V is such that either V ∩Mw = {0} or V → Mw is onto, then Gk andG′k intersect transversely at V .

Proof. We just remark that the proof of [B1, Proposition 3.2] goes throughin this more general set-up also. �

Corollary 3.5. R −R0 is nonsingular.

Proof. In the above notations, take

M = k2g+2,Mw0= D,Mwi

= Di for i > 0.

Then S = G0 ∩ G′0 and, if V ∈ S − S0, then V satisfies the conditions V ∩Mw = 0

for all w. Proposition 3.4 implies that R−R0 ≈ S − S0 is nonsingular. �

4. The moduli spaces

We keep the notations of sections 2 and 3. Fix a line bundle α of degree 2g+1with i-action onX; note that α is a square root ofOX(2g+1). Let F be a semistableorthogonal bundle of rank 4 with i-action. We regard F as a vector bundle witha nondegenerate quadratic form with values in OX . Let E = F ⊗ α, then E is avector bundle of rank 4 and slope 2g+1 with a nondegenerate quadratic form withvalues in OX(2g + 1). Our aim in this section is to prove the main theorem in thefollowing form.

Theorem 4.1 (Theorem 1.1). Let M be the moduli space of semistable vectorbundles E of rank 4 and slope 2g + 1 on the nodal hyperelliptic curve X with anondegenerate quadratic form with values in OX(2g+1) and with an i-action suchthat rw(E) = 1 for w �= w0 and rw0

(E) = 2. Then M is isomorphic to the quotientof R −R0 by (Z/2Z)2g.

The proof of the theorem is on similar lines to those of the main theorems in[B1], [B2], so we omit some details of the proofs. Most of the modifications neededin the nodal curve case have been worked out in sections 2 and 3. In the case whenthe curve is nonsingular, every ramification point w determines (uniquely) a Cartierdivisor. In the nodal case, we have to choose a suitable Cartier divisor Y at thenode w0 with OY ≈ OP1/m2

w0(section 2.1). We shall often use the same notation

for a ramification point in X and its image in P1. Let W = Y +

2g∑i=1

wi; thus W is

a Cartier divisor on X with i-action.Let E ∈ M. Recall that a superscript ‘−’ denotes the eigenspace corresponding

to the eigenvalue (−1) for the i-action. Consider the evaluation map

h : H0(E)− → (E ⊗OW )−.

48


The kernel of this map is contained in H0(E(−W ))−. Since E(−W ) is semistableand χ(E(−W )) = (−4g), by Lemma 2.1(b) we have H0(E(−W )) = 0. Thus his an injection. Recall that in Remark 3.1 we had the orthogonal decomposition

k2g+2 = D ⊕( 2g⊕

i=1

Di

)with respect to the pencil (I) as well as the dual pencil.

The quadratic form on E induces a quadratic map q0 on (E ⊗ OY )− with values

in OX(2g + 1) ⊗ OY ≈ OY . On D we have the quadratic map A0 with values inOP1/m2

w ≈ OY . Let n0 denote an isomorphism from (E ⊗ OY )− onto D which

preserves q0 on the left and A0 on the right. Let For i > 0, let ni denote anisomorphism from E−

wito Di which preserves qwi

on the left and Ai on the right.

Let n =

2g⊕i=0

ni. The composite n ◦ h embeds H0(E)− in k2g+2. Given (E, n), let

f(E, n) denote the image of H0(E)− in k2g+2 under n ◦ h. Since χ(E) = 4(g + 2),by Lemma 2.1(a), h1(E) = 0. As rw0

(E) = 2, rw(E) = 1, w �= w0, by Proposition2.3, we have h0(E)− = g + 3. It can be seen that E → f(E, n) modulo (Z/2Z)2g

gives a morphism from the moduli space M to the quotient by (Z/2Z)2g of theGrassmannian of (g + 3) dimensional subspaces of k2g+2. We shall see below thatthis morphism in fact maps into R−R0/(Z/2Z)

2g as f(E, n) ∈ R−R0.

Lemma 4.2. (a) H0(E(−x − ix))− is the orthogonal complement of H0(E)−

with respect to Qx for all x in X − (w0, · · · , w2g).In particular, H0(E(−x − ix))− is totally isotropic and Qx has rank 4 on

H0(E)−. Thus f(E, n) ∈ R.(b) f(E, n) �∈ R0.

Proof. (a) This follows as in [B1, Proposition 2.1 (b) and (c)], in fact the proofis slightly simpler in our case. In the notations of [B1],

∑Cw = k2g+2, h = O(1).

In our case, H1(E(−x − ix))− = 0 as χ(E(−1)) > 4(g − 1) (Lemma 2.1(a)), soh0(E(−1))− can be computed using Proposition 2.3.(b) In view of section 3.2, it suffices to show that

dim(f(E, n) ∩

2g⊕i=1

Di

)= g + 1.

Now the L.H.S. is isomorphic to the space H0(E ⊗mw0)−, where mw0

denotes theideal sheaf of the point w0. Using the facts that χ(E ⊗mw0

) = 4g, rw(E ⊗mw0) =

rw(E) = 1, w �= w0, and rw0(E ⊗mw0

) = 2, Lemma 2.1 and Proposition 2.3 giveh0(E ⊗mw0

)− = g + 1. �

4.1. The inverse of f . We now want to give the inverse of f . We define amorphism X×k2g+2 → OW by the conditions that it is zero outside W , its restric-tion to wi is given by mapping Di isomorphically onto k(wi) and the restriction tow0 is given by mapping D isomorphically onto OY . Let K be the kernel of thismorphism. Since W is a Cartier divisor and X is Cohen-Macaulay, K is locallyfree of rank 2g + 2. Since Q | K vanishes identically on W , it induces a quadraticform q on K with values in OX(2g+1− 2W ) = OX(−1) which is easily seen to beeverywhere nondegenerate.

49

8 USHA N. BHOSLE

Now take V ∈ R−R0. As seen in section 3.2, we have

V +(⊕

j �=i

Dj

)= k2g+2, i = 0, 1, · · · , 2g.

This implies that the composite X × V → X × k2g+2 → OW is a surjection. Thekernel V ′ of this composite is a vector subbundle of K fitting in the followingcommutative diagram:

0 → K → X × k2g+2 → OW → 0

↑ ↑ ‖0 → V ′ → X × V → OW → 0.

The orthogonal complement V ′′ of V ′ in K is a vector bundle of rank g − 1. SinceV ≈ V ′ generically, the rank of the induced form q on V ′ is ≤ 4 at any pointx. On the other hand dimV ′

x ∩ V′′

x ≤ g − 1 at every point x, so that the rank ofq | V ′

x ≥ 4. Thus q has constant rank 4 on V ′, V ′′ = V ′∩V ′′ and F ′ = V ′/V ′′ has anondegenerate quadratic form with values in OX(−1). Define E = F ′(g + 1); thusE is a vector bundle of rank 4 with a nondegenerate quadratic form with values inOX(2g + 1). The isomorphisms (ni) are obtained from the fact that V ′

wicontains

a copy of Di for all i �= 0 and V ′w0

contains a copy of D. We shall show below thatE is semistable. Define f ′(V ) = (E, n).

Lemma 4.3. Let E be as in section 4.1. Then H1(E)− = 0.

Proof. Using Serre duality and the fact that E ≈ E∗(2g + 1), we have

h1(E)− = h0(E∗(−2g − 1)⊗ ωX)− = h0(F ′(−g)⊗ ωX)−.

Since i acts on ωX by −Id at all w, there is an isomorphism ωX→OX(g − 1) notcompatible with the i-action. Consequently, h0(F ′(−g) ⊗ ωX)− = h0(F ′(−1))+

and it suffices to show that h0(F ′(−1))+ = 0. Now, H0(F ′(−1))+ ⊂ H0(F ′)+

and we shall in fact show that h0(F ′)+ = 0. From the cohomology exact sequenceassociated with the exact sequence

0 → V ′′ → V ′ → F ′ → 0,

we have h0(F ′)+ = 0 if h0(V ′)+ = 0 = h1(V ′′)+. One has H0(V ′) ⊂ H0(K) = 0 asH0(X × k2g+2) ≈ H0(OW ) implies that H0(K) = 0. Then h0(V ′) = 0 = h0(V ′′).As degF ′ = −4 and deg V ′ = −(2g + 2), one has deg V ′′ = −2g + 2 and r(V ′′) =g − 1. Hence by the Riemann-Roch theorem, h1(V ′′) = −χ(V ′′) = g2 − 1. Now

h1(V ′′)+ = −χ(V ′′)+ = 0 if and only if χ(V ′′)− = χ(V′′) = 1 − g2. Note that

V′−w = ker(V ′

w → (X × V )w) has a nondegenerate quadratic form at w while V ′′w

is isotropic for all w. Hence rw(V′′) = 0 for all w. Then Proposition 2.3 gives

χ(V ′′)− = 1− g2. This proves the lemma. �

Lemma 4.4. There is a canonical isomorphism of V onto H0(E)− such thatthe following diagram commutes:

V → k2g+2

↓ || nH0(E)− → (E ⊗OW )−.

Proof. This follows as in [B1, Lemma 2.4]. �

50


Lemma 4.5. Let E be as in section 4.1. Then F = E ⊗ α−1 is a semistableorthogonal bundle with i-action.

Proof. Let L0 ⊂ F be an i-invariant isotropic subsheaf with a torsionfreequotient, L = L0 ⊗ α ⊂ E. We identify (E ⊗ OW )− with k2g+2 by n. Since L isisotropic, L−

wi⊂ Di, i �= 0, and Im (L ⊗ OY )

− ⊂ D are isotropic. Hence L−wi

= 0for i �= 0 and Im(L⊗OY )

− ⊂ D0. Thus Im (L⊗OW )− ⊂ D0. Now,

H0(L)− ⊂ Im (L⊗OW )− ∩H0(E)− ⊂ D0 ∩H0(E)− ⊂ D0.

Hence h0(L)− ≤ 1.If D0 ∩ H0(E)− = {0}, then H0(L)− = 0. Since H0(L⊥)− ⊂ H0(E)− and

h1(E)− = 0 (Lemma 4.3), we have h0(L⊥)− ≤ χ(E)−. Hence

h0(L)− + h0(L⊥)− ≤ χ(E)−

and so χ(L)− + χ(L⊥)− ≤ χ(E)−. Using Proposition 2.3, this implies that

d(L0) + d(L⊥0 ) ≤ d(F ).

Suppose that D0 ∩H0(E)− �= {0}. If dim (L ⊗ OY )− = rw0

(L) = 0, then westill have H0(L)− = {0} and the previous argument goes through. Assume thatrw0

(L) �= 0. Since

0 → L⊥ → E → L∗ ⊗OX(2g + 1) → 0

is exact, rw0(L) + rw0

(L⊥) = rw0(E) = 2 and, L ⊂ L⊥ being isotropic, one must

have rw0(L) = rw0

(L⊥) = 1, i.e. Im(L⊥ ⊗OY )− = D0. Then

H0(L⊥)− ⊂(D0 ⊕

(⊕i>0

Di

))∩H0(E)− =

(D0 ⊕

(⊕i>0

Di

))∩ V

by Lemma 4.4. Since V ∈ R−R0, we have V +(D0 ⊕

(⊕i>0

Di

))= k2g+2 (section

3.2). Hence

dim V ∩(D0 ⊕

(⊕i>0

Di

))= dim V − 1.

We already have h0(L)− ≤ 1. Thus h0(L)− + h0(L⊥)− ≤ dim V = χ(E)−, so thatχ(L)−+χ(L⊥)− ≤ χ(E)− in this case also and hence degree L0+ degree L⊥

0 ≤ degF . This proves that F is a semistable orthogonal bundle with i-action. �

Corollary 4.6. The canonical map H0(E)− → (E ⊗ OW )− is an injectionand f(E, n) = V. Thus f ◦ f ′ = Id.

Proof. This is obvious from Lemma 4.4. �

Lemma 4.7.

f ′ ◦ f = Id.

Proof. This follows from f◦f ′ = Id and the fact that two semistable quotientsof the same rank and degree of the trivial (fixed rank) bundle, which are genericallyisomorphic, are isomorphic. �

This finishes the proof of Theorem 1.1.

51

10 USHA N. BHOSLE

Remark 4.8. We can use Theorem 1.1 to construct a birational map betweenUL2 and R exactly as in [R, § 6]. To prove an analogue of [R, Theorem 2, § 6], we

need Lemma 2.2 and Remark 3.3 together with the fact that H2(X,O∗) = 0 for thenodal curve X too. Notice also that the orthogonal bundle with i-action associatedwith E is E ⊗ i∗E in [R, § 6.3], the i-action being swiching factors E, i∗E, and(E ⊗ i∗E ⊗OY )

− ≈ (Λ2(E)⊗OY )−; hence rwo

(E ⊗ i∗E) = 2, rw(E ⊗ i∗E) = 1 forw �= w0.

Note that the tensor product of two stable vector bundles E and i∗E on thenodal curve X may not be semistable. If the pull back E′ of E to the normalizationX is stable, then E′⊗i∗E′ is semistable (as X is smooth). Since E⊗i∗E is a vectorbundle, the semistability of E′ ⊗ i∗E′ implies the semistablity of the generalizedparabolic bundle corresponding to E ⊗ i∗E, hence E ⊗ i∗E is semistable. Thusthere is a nonempty open subset U of the moduli space UL

2 such that for E ∈U, E ⊗ i∗E ∈ M. Hence we get only birationality rather than an isomorphism asin [R].

References

[B1] Bhosle Usha N. Moduli of orthogonal and spin bundles over hyperelliptic curves, Compo-sitio Math. 51 (1984), no.1, 15–40.

[B2] Bhosle Usha N. Degenerate symplectic and orthogonal bundles on P 1, Math. Ann. 267(1984), no.3, 347–364.

[B3] Bhosle Usha N. Pencils of quadrics and hyperelliptic curves in characteristic two, J. ReineAngew. Math. 407 (1990), 75–98.

[B4] Bhosle Usha N. Vector bundles of rank 2, degree 0 on a nodal hyperelliptic curve, Algebraicgeometry (Catania, 1993/Barcelona, 1994), 271–281, Lecture notes in Pure and Appl.Math., 200, Dekker, New York, 1998.

[B5] Bhosle Usha N. Moduli of vector bundles in characteristic 2, Math. Nachr. 254/255 (2003),11–26.

[DR] Desale U.V. and Ramanan S. Classification of vector bundles of rank 2 on hyperellipticcurves, Invent. Math. 38 (1976/77), no.2, 161–185.

[NRa] Narasimhan M.S., Ramanan S. Moduli of vector bundles on a compact Riemann surface,Ann. of Math. (2) 89 (1969), 14–51.

[N] Newstead P.E. Stable bundles of rank 2 and odd degree over a curve of genus 2, Topology7 (1968), 205–215.

[R] Ramanan, S. Orthogonal and spin bundles over hyperelliptic curves, Geometry and anal-ysis (papers dedicated to the memory of V.K. Patodi), pp. 151–166, Indian Acad. Sci.,Bangalore, 1980.




52


Hilbert Schemes of Fat r-Planes and the Triviality ofChow Groups of Complete Intersections

Andre Hirschowitz and Jaya NN Iyer

Abstract. In this paper, we investigate the question of triviality of the ra-tional Chow groups of complete intersections in projective spaces and obtainimproved bounds for this triviality to hold. Along the way, we have to studythe dimension and nonemptiness of some Hilbert schemes of fat r-planes con-tained in a complete intersection Y , generalizing well-known results on theFano varieties of r-planes contained in Y .

1. Introduction

1.1. The triviality conjecture. The aim of this paper is to investigate thetriviality of the low-dimensional rational Chow groups for certain projective va-rieties. If Y is a nonsingular complete intersection of multidegree (d1, · · · , ds) ina projective space Pn, and n is sufficiently large with respect to the degrees, it isknown that, for small values of r, the rational Chow group QCHr(Y ) := CHr(Y )⊗Q

is trivial, namely one-dimensional (generated by the linear sections). The preciseconjectural bound on the multidegrees for the triviality follows from the study ofthe Hodge type of the complementary open variety Pn−Y initiated by Deligne [De]and followed by works of Deligne-Dimca [De-Di] and Esnault-Nori-Srinivas [Es],[EsNS]. A formulation of the conjectured bound was made in [Pa, Conjecture 1.9],which says:

Conjecture 1.1. Suppose Y ⊂ Pn is a smooth complete intersection of mul-tidegree (d1, · · · , ds), and let r be a nonnegative integer. If

r(max1≤i≤sdi) +

s∑i=1

di ≤ n,

then QCHr(Y ) is trivial.

This conjecture includes the hypersurface case (s = 1) as well as the higher-dimensional case s ≥ 2, and, as we will see, our contribution concerns mostly thelatter.

2010 Mathematics Subject Classification. Primary 14C25, 14D07; Secondary 14D22,14F40.Key words and phrases. Complete intersections, Chow groups, Hilbert schemes.


1



53

2 A.HIRSCHOWITZ AND J. N. IYER

1.2. Fat and strong planes. The central notions in our approach are thoseof fat and strong planes, which appear at least implicitly in [EsLV], and go backto Roitman for the 0-dimensional case.

By a t-fat r-plane in a projective space, we mean the t-th infinitesimal neigh-borhood of an r-plane in an (r + 1)-plane. Given a subscheme Y ′ in a projectivespace and a Cartier divisor Y in Y ′, we say that an r-plane L in Y is strong (withrespect to Y ′) if there exists an (r + 1)-plane L′ in Y ′ containing L such that theset-theoretic intersection L′ ∩ Y is either L or L′. The connection between the twonotions is given by the following statement, proven in section 2 :

Proposition 1.2. Suppose Y is in the linear system |OY ′(t)|. Then any strongr-plane in Y is the support of a t-fat r-plane contained in Y . Conversely, if fur-thermore Y ′ is (set-theoretically) defined by equations of degree (strictly) less thant, then the support of any t-fat r-plane contained in Y is strong.

1.3. Roitman’s technique, small steps and big steps. The case of 0-cycles has been handled by Roitman [Ro]. His method consists in starting froma (positive) 0-cycle Z on Y , and building a ruled cycle in Pn whose intersectionwith Y will be not too far from a multiple of Z. This is achieved by choosing aruling by lines which are strong (see below), in the sense that either they cut Yin a single (multiple) point or they are inside Y . This method can be extendedto the higher-dimensional case of r-cycles. Of course, the scope of this method islimited by the need for “sufficiently many strong (r + 1)-planes”. This approachhas been successfully applied in [EsLV] through a single big step, showing that therestriction of (r+s)-cycles from P

n to Y is sufficiently surjective under the followingnumerical assumption (at least for degrees at least 3, the assumption being differentwhen all degrees are equal to 2):

n ≥s∑

i=1

(di + r

r + 1

).

The geometric meaning of their numerical condition is the rational-connectednessof the variety of r-planes in Y .

In the present work, we apply Roitman’s technique through smaller steps, typi-cally showing that the restriction of (r+1)-cycles on a suitable complete intersectionof multidegree (d1, · · · , ds−1) to Y is sufficiently surjective. The analysis of smallsteps being somewhat simpler, we succeed in applying Roitman’s technique to smallsteps in essentially the whole expected range, relaxing in particular the rational-connectedness assumption. For instance for 5-cycles on complete intersections oftype (20, 30), the rational-connectedness condition requires n ≥ 1800000 while wetake care of all cases with n ≥ 370000.

1.4. Our small step theorem. Our first main result reads as follows:

Theorem 1.3. If, in the subvariety Y ′ ⊂ Pn, the Cartier divisor Y ∈ |OY ′(t)|is covered by strong r-planes, then the restriction map QCHr+1(Y

′) → QCHr(Y )is onto.

This implies in particular that whenever QCHr+1(Y′) is trivial, so is QCHr(Y ).

Our proof of Theorem 1.3 follows the corresponding proof for the big step in[EsLV], with a single but decisive technical improvement: we introduce a different

filtration of the Chow group CHr, where CH(s)r is generated by subvarieties covered

A. HIRSCHOWITZ AND J. NN IYER54

HILBERT SCHEMES OF FAT r-PLANES 3

by strong s-planes. We did not explore yet whether such a new filtration couldalso improve the bound for the big step.

In order to apply the above result, we need to find the appropriate conditionon the degrees for our complete intersection Y to be covered by strong r-planes.

1.5. Covering by strong planes. Thus we are led to search for the numericalcondition for at least the generic complete intersection of multidegree (d1, · · · , ds)to be covered by strong r-planes.

We recall that the strongness property is with respect to the pair (Y, Y ′). Wesay that an s-codimensional subvariety in a projective space has type (d1, · · · , ds)when it is a union of irreducible components of a complete intersection of multi-degree (d1, · · · , ds). Accordingly, we say that a pair Y ⊂ Y ′ has type (d1, · · · , ds)if Y ′ has type (d1, · · · , ds−1) and Y is a divisor of degree ds in Y ′. In §6, we prove

Proposition 1.4. Let n, r, s, d1, · · · , ds be integers satisfying

r ≥ 0, s ≥ 1, n ≥ r + s, 2 ≤ d1 ≤ · · · ≤ ds−1 < ds

and the (“expected”) inequalityρ+ r ≥ n− s,

where

ρ := (r + 2)(n− r)−s∑

i=1

(di + r + 1

r + 1

)

is the dimension of the variety of ds-fat r-planes in the general complete intersectionof type (d1, · · · , ds) (see §4).

If Y ⊂ Y ′ is any pair of type (d1, · · · , ds) in Pn, then Y is covered by strongr-planes.

Note that the intended meaning of the “expected” inequality relates the dimen-sion of the universal ds-fat r-plane with the dimension of our complete intersectionY .

Also note the strict inequality ds−1 < ds. Apart from this restriction, our resultis the expected one. The discarded case would involve a refined analysis (this iswhere we do not cover the whole range of Roitman’s method for small steps).

Our proof of Proposition 1.4 relies on the study of the Hilbert schemes of fat r-planes contained in a general complete intersection. We show in §5 that they havethe expected dimension; but we need a more accurate result saying that, whenthis expected dimension is nonnegative, these Hilbert schemes are nonempty. Weconjecture that this is true in most cases (despite the notable exception of doublelines on quadric surfaces), and prove it in the case we need for our application toChow groups. For such a result, as illustrated in [De-Ma], two approaches areavailable: through intersection computations or through maximal rank problems.We follow the latter approach, using a method that can be tracked back at least to[EH, EHM].

1.6. The main theorem. Combining the previous results, we obtain ourmain result:

Theorem 1.5. Let n, r, s, d1 ≤ · · · ≤ ds−1 < ds be integers as above, satisfying

ρ+ r ≥ n− s.

If Y ⊂ Y ′ is any pair of type (d1, · · · , ds) in Pn, and if QCHr+1(Y′) is trivial,

then so is QCHr(Y ).

55


This theorem may be applied recursively. For instance in the case of codimen-sion two complete intersections (s = 2), our assumption for triviality reads, ford1 < d2 :

n ≥ max

{(d1+r+1

r+1

)+(d2+r+1

r+1

)+ r2 + r − 2

r + 1,

(d1+r+2

r+2

)+ r2 + 3r

r + 2

}.

In order to compare this new bound with [EsLV]’s

n ≥(d1 + r

r + 1

)+

(d2 + r

r + 1

),

we fix r and d1 and let d2 vary. For sufficiently large values of d2, the new bound

can be estimated asdr+12

(r+1)(r+1)! , while, if one uses [EsLV], the best value of n is

estimated asdr+12

(r+1)! . Hence in this context of a large d2, we roughly divide by r+1

the range where the conjecture is still open. On the other hand, it can be checkedthat, if d2 is not sufficiently large, [EsLV]’s bound remains the best (for instancefix d2 = d1 + 1 and let d1 go to infinity).

In the higher-codimensional cases, a similar picture will occur, namely ourresult will provide an improved bound only for sufficiently large values of ds. Fur-thermore, in many cases, the best bound will be obtained by combining one or moreof our small steps with a big step from [EsLV].

1.7. The case of hypersurfaces. The case of hypersurfaces (s = 1) has beenconsidered in the first place. Concerning a general cubic hypersurface Y ⊂ Pn, C.Schoen [Sc] showed the triviality QCH1(Y ) Q when n ≥ 7 and Paranjape [Pa]obtained the sharp bound in this case showing the triviality of 1-cycles when n ≥ 6(in the same paper, he gave the first finite bound for general complete intersections).

For hypersurfaces of the general degree d, the best known bound has beenobtained “in the margin” by J. Lewis [Le] (added in proof at the very end of thepaper). There, the statement concerns only the generic hypersurface, and the boundoccurs as the condition for the so-called cylinder homomorphism to be surjective.This bound by Lewis is better than the bound obtained later (for the hypersurfacecase) in [EsLV]. It was rediscovered by A. Otwinowska [Ot]: there the statementconcerns all smooth hypersurfaces, and the geometric meaning of the bound is thatthe hypersurfaces of degree d in Pn+1 (not Pn!) are covered by (r + 1)-planes.Surprisingly, our small step gives exactly the same bound, with a third geometricmeaning for the condition, namely that the hypersurfaces are covered by d-fat r-planes. Furthermore, our statement concerns all hypersurfaces, not only smoothones.

1.8. The base field. We work over an algebraically closed field of character-istic zero. The closedness assumption could be removed, thanks to the fact thatthe kernel of CHr(Yk) → CHr(Yk) is torsion [Bl], while the characteristic zeroassumption is used in the proofs of §4.

Acknowledgements : It is a pleasure to dedicate this paper in honour of S.Ramanan. Both authors have experienced very fruitful mathematical interactionwith him, and take the opportunity to acknowledge his deep influence. This workwas initiated during the second author’s stay at MPI, Bonn in 2003 and partly doneduring her visits to Nice in Dec 2004 and to IAS, Princeton in 2007. The support



and hospitality of these institutions is gratefully acknowledged. We also thank H.Esnault, J. Lewis and M. Levine for their feedback at some point during the courseof this work.

2. Strong planes

Throughout this section, we consider a subvariety Y ′ in a projective spaceequipped with a Cartier divisor Y , and we fix an integer r. We are interested inthe restriction map QCHr+1(Y

′) → QCHr(Y ), where we write QCHr(W ) for therational Chow group of r-dimensional cycles on W .

Recall that an r-plane L in Y is said to be strong (with respect to Y ′) if thereexists an (r+1)-plane L′ in Y ′ containing L such that the set-theoretic intersectionL′ ∩ Y is either L or L′.

In this section, we prove our first main result :

Theorem 2.1. If, in the subvariety Y ′ ⊂ Pn, the Cartier divisor Y ∈ |OY ′(d)|is covered by strong r-planes, then the restriction map QCHr+1(Y

′) → QCHr(Y )is onto.

For the proof, we generalize our notion of strongness and define a notion ofstrong s-plane in Y ′ for s ≤ r+1. A (r+1)-plane H in Y ′ is said to be strong (withrespect to the pair (Y, Y ′)) if it is contained in Y , or if its set-theoretic intersectionwith Y is a r-plane. Then, for s ≤ r, a s-plane in Y ′ is said to be r-strong, or simplystrong (when r is clear from the context), if it is contained in a strong (r+1)-plane.As usual, we say that a closed subvariety W of Y is spanned or covered by strongs-planes if it is a union of strong s-planes contained in W .

Now we denote by QCH(s)r (Y ) the subgroup of QCHr(Y ) which is generated by

r-dimensional subvarieties of Y which are spanned by strong s-planes. This is theplace where our proof differs from the corresponding proof in [EsLV]. Note thatany subvariety in Y is spanned at least by strong 0-planes: since Y is covered by

strong r-planes, it is also covered by strong 0-planes. Thus we have QCH(0)r (Y ) =

QCHr(Y ).For s ≥ 1, if Z is spanned by strong s-planes it is spanned by strong (s − 1)-

planes as well. Hence one has QCH(s)r (Y ) ⊆ QCH(s−1)

r (Y ). For s > r one has

QCH(s)r (Y ) = {0}.We prove by descending induction on s that QCH(s)

r (Y ) is in the image of

QCHr+1(Y′). The initial case is with s := r + 1 and follows since QCH(r+1)

r (Y ) isreduced to 0.

Before stating the induction step as a lemma, we introduce the following nota-tion. Let Γ ⊂ Y ′ be an (r+1)-dimensional closed subvariety or, more generally, an(r+1)-cycle. By [Fu, 8.1], the intersection product Γ ·Y is a class in CHr(|Γ| ∩Y ).By abuse of notation we will also write Γ · Y for its image in QCHr(Y ).

Lemma 2.2. Let s be an integer with 0 ≤ s ≤ r, and W an r-dimensionalirreducible subvariety of Y , spanned by strong s-planes but not by strong (s + 1)-planes. Then there exist an (r+1)-dimensional cycle Γ in Y ′ and a positive integerα with

Γ · Y ≡ αW mod QCH(s+1)r (Y ).

57


Proof. We start with the case s = r which means that W is a strong r-plane.This gives us a strong (r + 1)-plane in Y ′ which we take for Γ. Indeed, we haveΓ · Y = dW.

Now we suppose s < r. In order to define Γ, we start by choosing carefullyan algebraic family (Hz)z∈Z of strong s-planes covering W . Note that by ourassumption on s, each strong s-plane in Y is contained in a strong (s + 1)-planealso contained in Y , thus we may choose more precisely an algebraic family (Hz ⊂H ′

z)z∈Z where H ′z is a strong (s + 1)-plane in Y , Hz is a hyperplane in H ′

z andW is covered by (Hz)z∈Z . By standard arguments, we may suppose that Z isprojective smooth connected of dimension r − s. We denote by HZ ⊂ H ′

Z the twocorresponding projective bundles over Z.

Since W is not covered by strong (s + 1)-planes, the projection of H ′Z into

Y is not contained in W , thus it is a positive (r + 1)-cycle. We take for Γ thisChow-theoretic projection of H ′

Z in Y ′.Let us now compute Γ ·Y in QCHr(Y ) (remember that we consider Γ as a cycle

in Y ′).We start by applying the projection formula [Fu, 8.1.7] to pr2 : Z × Y ′ → Y ′:

Γ · Y = pr2∗(H′Z) · Y = pr2∗(H

′Z · (Z × Y )).

So now we compute H ′Z · (Z × Y ). This is the divisor class in H ′

Z defined by thelinear system |pr∗2OY ′(Y )|. Now H ′

Z is a projective bundle and this linear systemhas degree d along the fibers of this bundle. Thus we have

H ′Z · (Z × Y ) = dHZ + ψ−1(D),

where D is a divisor in Z and ψ : H ′Z → Z is the bundle projection. We get

Γ · Y = dpr2∗(HZ) + pr2∗ψ−1(D)

in QCHr(Y ∩ pr2(H′Z)). Since HZ is generically finite over the subvariety W and

since pr2∗(ψ−1(D)) lies in QCH(s+1)

r (Y ), one obtains, for some positive multiple αof d, the relation

Γ · Y ≡ αW mod QCH(s+1)r (Y ).

�

Now we check the following statement, already mentioned in our introduction:

Proposition 2.3. Suppose Y is in the linear system |OY ′(t)|. Then any strongr-plane in Y is the support of a t-fat r-plane contained in Y . Conversely, if fur-thermore Y ′ is (set-theoretically) defined by equations of degree (strictly) less thant, then the support of any t-fat r-plane contained in Y is strong.

Proof. For the first statement, our strong r-plane L is contained in a strong(r + 1)-plane L′ ⊂ Y ′. If L′ is contained in Y , then so is the t-th infinitesimalneighborhood of L in L′. If not, then, since the set-theoretic intersection of Y andL′ is L, and the degree of the restriction of |OY ′(t)| to L′ is t, the scheme-theoreticintersection Y ∩ L′ has to be the t-th infinitesimal neighborhood of L in L′. Thusin both cases, this is the desired t-fat r-plane.

For the second statement, let L ⊂ Pn be a t-fat r-plane contained in the (r+1)-plane L′. The equations defining Y ′ vanish on L. Since these equations can be



chosen of degree strictly less than t, they vanish identically on L′, which meansthat L′ is contained in Y ′, hence that L is strong.

�

3. Restricted flag-Hilbert schemes

In this section, we collect some technical material concerning the infinitesimaltheory of restricted flag-Hilbert schemes. Here by a full Hilbert scheme (for a givenprojective variety), we mean any open subscheme of the Hilbert scheme associatedto a Hilbert polynomial, while by a Hilbert scheme, we mean any locally closedsubscheme of a full Hilbert scheme.

Given two full Hilbert schemes H1 and H2 of subschemes of the same ambientprojective scheme P , we have the corresponding flag-Hilbert scheme D of pairs(X ↪→ Y ) in H1 × H2. Two subschemes H′

1 ⊂ H1 and H′2 ⊂ H2 being given,

by the corresponding restricted flag-Hilbert scheme, we mean the scheme-theoreticintersection D′ of D with H′

1 ×H′2.

In the example we have in mind, P is a projective space, H′1 is a variety of fat

planes, and H′2 = H2 is a full Hilbert scheme of complete intersections.

We write i : X → Y for a given pair, IX and IY for the two ideal sheaves onP , NX := Hom(IX ,OX) and NY := Hom(IY ,OY ) for the corresponding normalbundles. We denote by NY |X the restriction i∗(NY ) of NY to X. We also havei∗ : NX → i∗NY and i∗ : NY → i∗NY |X . Note that the two codomains have the

same space of sections H0(NY |X). Putting this together, we have a morphism

(i∗, i∗) : H0(NX)⊕H0(NY ) → H0(NY |X).

The domain of this morphism is the tangent space to the product of our two Hilbertschemes, and the tangent space to the flag-Hilbert scheme is identified as the kernelof the above map (i∗, i

∗) (see [Kl], [Se, Remark 4.5.4 ii]). Hence the differentialsof the two projections are the restrictions to this kernel of the projections.

We first state in our way the standard result in the unrestricted case:

Proposition 3.1. We suppose that H1, H2 are smooth connected and that Dhas codimension c at O := (X,Y ). We also suppose that i∗ : H0(NY ) → H0(NY |X)has rank c. Then

(i) D is smooth at O;(ii) the image of i∗ : H0(NX) → H0(NY |X) is contained in the image of i∗ :

H0(NY ) → H0(NY |X);(iii) the first projection D → H1 is smooth at (X,Y );(iv) the second projection D → H2 is smooth at (X,Y ) if (and only if) the rank

of i∗ : H0(NX) → H0(NY |X) is c.

Proof. (i) Since i∗ has rank c, the pair (i∗, i∗) has rank at least c. It follows

that, in the tangent space of H1 × H2 at O, the tangent space to D is at leastc-codimensional. Since D is c-codimensional, this implies that D is smooth at O.

(ii) By the previous argument, we see that the rank of the pair (i∗, i∗) is exactly

c, which means the stated inclusion.(iii) Using the previous item and an easy diagram chase, we see that the differ-

ential of D → H1 at O is onto.(iv) This follows by a similar diagram chase.

�

59


Now we turn to the restricted case. Here we write K0(NX) ⊂ H0(NX) for thetangent space to H′

1 at X, and K0(NY ) ⊂ H0(NY ) for the tangent space to H′2 at

Y .

Proposition 3.2. We suppose that H′1, H′

2 are smooth connected and that D′

has codimension c at O := (X,Y ). We also suppose that i∗ : K0(NY ) → H0(NY |X)has rank c. Then

(i) D′ is smooth at O;(ii) the image of i∗ : K0(NX) → H0(NY |X) is contained in the image of i∗ :

K0(NY ) → H0(NY |X);(iii) the first projection D′ → H′

1 is smooth at (X,Y );(iv) the second projection D′ → H′

2 is smooth at (X,Y ) if (and only if) therank of i∗ : K0(NX) → H0(NY |X) is c.

Proof. The main point is the identification of the tangent space to D′: a pair(t1, t2) of vectors in K0(NX)×K0(NY ) is tangent to D′ if the subscheme (over Speck[ε]) corresponding to t1 is included in the one corresponding to t2. This meansexactly that (t1, t2) is tangent to D. Hence the tangent space to D′ is the kernel ofthe restriction (i∗, i

∗) : K0(NX)⊕K0(NY ) → H0(NY |X). The rest of the proof isidentical to the previous one.

�

4. Fat planes in complete intersections

In this section, we consider

• a projective space Pn,

• an integer r with 0 ≤ r < n, which is the dimension of our (fat) planes,• an integer s with 1 ≤ s ≤ n − r − 1, which is the codimension of ourcomplete intersections (or the number of their equations),

• a sequence d := (d1, · · · , ds) of s positive integers, which is the multidegreeof our complete intersections,

• an integer t, with 2 ≤ t ≤ maxd, which is the multiplicity of our fatr-planes.

We keep the notations of the previous section for our case where H′1 is the

(smooth) Hilbert scheme parametrizing t-fat r-planes in Pn and H′2 = H2 is the

(smooth) Hilbert scheme of complete intersections of type d. We write δ′i for thedimension of H′

i. The dimension δ′1 of H′1 does not depend on t (thanks to the

assumption t ≥ 2), it is the dimension of the corresponding flag variety, namely(r + 2)(n− r − 1) + r + 1, in other words (r + 2)(n− r)− 1.

We set ρ := (r + 2)(n − r) − 1 − Σsi=1

(di+r+1r+1

)+ Σdi≥t

(di−t+r+1

r+1

). We will

see that ρ is the expected dimension for the Hilbert scheme of t-fat r-planes in acomplete intersection of type d in P

n. Recall that by a t-fat r-plane, we mean thet-th infinitesimal neighborhood of an r-plane in an (r + 1)-plane. Finally, we setc := δ′1−ρ. Hence we have ρ = δ′1− c which means that c is the (expected) numberof conditions imposed on a t-fat r-plane for it to be contained in a given completeintersection of type d. The first result of this section confirms this expectation.



Proposition 4.1. (i) The codimension of the restricted flag-Hilbert scheme D′

in H′1 ×H′

2 is c.(ii) For the generic complete intersection Y of type d in P

n the dimension ofthe Hilbert scheme of t-fat r-planes in Y is everywhere ρ. In particular this Hilbertscheme is empty if ρ is negative.

Proof. (i) We consider a variety V parameterizing our complete intersections,

namely the open subset of the vector space V of s-tuples of homogeneous polyno-mials (in n+1 variables) of the given multidegree defining an s-codimensional sub-scheme in P

n. We write dV for the dimension of V . The variety V comes equippedwith the tautological subscheme T ⊂ V ×P

n. The corresponding morphism V → H2

is surjective and it is easily checked to be smooth. Similarly, H′1 comes equipped

(thanks to t ≥ 2) with a tautological flag L ⊂ L′ ⊂ H′1 × Pn, where L is the tauto-

logical t-fat r-plane, while L′ is its linear span: its fibers over H′1 are (r+1)-planes.

Next, we introduce the incidence subscheme D := D′ ×H2V ⊂ H′

1 × V . SinceV → H2 is surjective and smooth, it is enough to prove that the codimension of Din H′

1 × V is c.The dimension dD of D is understood through the projection on H′

1. Indeed

the fibers of the projection D → H′1 are traces on V of sub-vectorspaces in V .

So we have to compute the codimension in V of tuples vanishing on a fixed t-fatr-plane L. This codimension is Σs

i=1ci, where ci is the codimension of homogeneous

polynomials of degree di vanishing on L. We easily check ci =(di+r+1r+1

)−(di−t+r+1

r+1

),

where we adopt the convention that(pq

)is zero whenever p < q. Hence we end up

with the desired result

dD = dV + ρ.

(ii) This is an immediate consequence of the first item.�

We need a complementary statement which is a particular case of the followingconjecture:

Conjecture 4.2. Apart from the exception below, for the generic completeintersection Y of type d in Pn, when ρ is nonnegative, the Hilbert scheme of t-fatr-planes in Y is nonempty.

Here is the known exception :

Example 4.3. For double lines on the generic quadric in P3, we have ρ = 0while the corresponding Hilbert scheme is empty.

In the rest of this section, we reduce the above conjecture to a maximal rankproblem. This maximal rank problem for the particular case we need will be handledin the next section.

We want to apply the result of the previous section. So we start from a flagH ⊂ L ⊂ L′ ⊂ Pn where L is a t-fat r-plane with support H and linear spanL′. Our first task is to identify the tangent space TLH′

1 at L with the varietyH′

1 of fat planes. Recall that the tangent space at L to the full Hilbert schemeis H0(L,NL), where NL := Hom(IL,OL) is the normal bundle. Hence we lookfor a subspace of that vector space. We choose coordinates xi where L is definedby the equations xt

0 = x1 = · · · = xn−r−1 = 0 so that we may identify NL as

61


the direct sum OL(1)n−r−1 ⊕ OL(t) and accordingly H0(NL) as the direct sum

H0(OL(1))n−r−1 ⊕H0(OL(t)).

For the following lemma, we will introduce again a notation K0. The readershould be aware that, in the present section, this notation is introduced in such away thatK0 differs fromH0 only in the special case where r is zero. For each integera, we denote by K0(OL(a)) the image of the restriction H0(OPn(a)) → H0(OL(a)).We also extend this notation to sequences in the natural way: by OL(d) we mean⊕iOL(di), and K0(OL(d)) stands for ⊕iK

0(OL(di)). Finally we set p := n− r− 1.

Lemma 4.4. (i) The image of the natural morphism j : TPn → NL fromthe tangent sheaf of Pn to the normal sheaf NL is a subsheaf N ′

L isomorphic toOL(1)

p ⊕OH(1) as an OL-module.(ii) More precisely, we may choose an isomorphism between NL and OL(1)

p ⊕OL(t) so that the corresponding injection OL(1)

p⊕OH(1) → OL(1)p⊕OL(t) decom-

poses as ι⊕μ, where ι is the identity on the first summand and μ is the multiplicationby xt−1

0 on the second one.(iii) The tangent space TLH′

1 at L to the variety H′1 of fat planes is the image

of H0(TPn) (or H0(OPn(1)n+1)) in H0(N ′L) (or H0(NL)). We write K0(N ′

L) forthis image.

(iv) Under the identification in (i), K0(N ′L) becomes K0(OL(1))

p⊕H0(OH(1)).

Proof. Let us start with the third statement. Since H′1 is the orbit in the full

Hilbert scheme of L under the projective linear group, TLH′1 has to be the image

of the natural map H0(j) : H0(TPn) → H0(NL).Now we turn to (i) and (ii). Using our coordinates, our morphism j, viewed as

a morphism from OPn(1)n+1 to OL(1)p ⊕OL(t), is given by the partial derivatives

of our n − r equations, which gives essentially the announced matrix: just notethat, thanks to the characteristic zero assumption, the image of the multiplicationby the partial derivative txt−1

0 from OL(1) to OL(t) is the same as the image of themultiplication by xt−1

0 , and this image is isomorphic to OH(1).Now we turn to (iv). We just note that by i), H0(N ′

L) is equal to H0(OL(1))p⊕

H0(OH(1)). By definition, K0(N ′L) is the image of H0(OPn(1)n+1) in H0(N ′

L),which can now be identified as the space K0(OL(1))

p ⊕H0(OH(1)).�

Now we consider a flag i : L → Y of complete intersection subschemes in Pn

where Y is the general complete intersection of type d containing L. The tangentspace at Y to the corresponding Hilbert scheme is H0(Y,NY ) which can be com-puted asH0(Y,OY (d)). It follows thatH

0(L,NY |L) is isomorphic toH0(L,OL(d)).

Thus we can write K0(L,NY |L) or simply K0(NY |L) for the image of H0(NY ) in

H0(L,NY |L).

Lemma 4.5. (i) For r > 0, K0(NY |L) is the whole of H0(L,NY |L).

(ii) In any case, the dimension of K0(NY |L) is c.

(iii) In any case, the natural map K0(N ′L) → H0(L,NY |L) factors through

K0(NY |L).

(iv) If the induced map K0(N ′L) → K0(NY |L) is onto, then D′ → H′

2 is smoothat (L, Y ).

Proof. (i) We know that NY is the direct sum OY (d) and a standard coho-mological argument shows that all its sections come from the ambient projective



space. Hence what we have to prove is that any section of OL(d) comes from theambient projective space, which follows from the standard cohomological argument:the cohomology of line bundles on projective spaces of dimension at least two istrivial (here we use r ≥ 1).

(ii) As we have just seen, K0(NY |L) is the image of H0(OPn(d)) in H0(OL(d)),so this is just a count of monomials which we leave to the reader.

(iii) and (iv) Now we apply Proposition 3.2: in our case, we have K0(NY ) =H0(NY ) and, according to Lemma 4.4, the assumption in Proposition 3.2 is pre-cisely the previous item. The statements (iii) and (iv) here are exactly the conclu-sions (ii) and (iv) of Proposition 3.2.

�

We turn to the final result of the present section where, for sake of clarity, wehandle separately the case r = 0. We will write 1,2 and t respectively for thesequence (1, · · · , 1), (2, · · · , 2) and (t, · · · , t), hence accordingly d− 1, d− 2, d− trespectively for (d1 − 1, · · · , ds − 1), (d1 − 2, · · · , ds − 2), (d1 − t, · · · , ds − t).

Proposition 4.6. (i) For r ≥ 1, we consider the generic morphismm : OL(1)

p ⊕OH(1) → OL(d) of coherent OL-modules. If

H0(m) : H0(OL(1)p)⊕H0(OH(1)) → H0(OL(d))

is onto, then D′ → H′2 is also onto.

(ii) For r = 0, we denote by K0(OL(1)p ⊕ OH(1),OL(d)) the image of the

natural map

K0(OL(d− 1))⊕K0(OL(d− t)) → Hom(OL(1)p ⊕OH(1),OL(d)).

We consider the generic morphism m in K0(OL(1)p⊕OH(1),OL(d)). If the image

by H0(m) of K0(OL(1)p)⊕H0(OH(1)) in H0(OL(d)) is K

0(OL(d)), then D′ → H′2

is onto.

Proof. (i) Since our morphism D′ → H′2 is projective and the codomain is

irreducible, it is sufficient to prove that it is dominant. We apply Lemma 4.5 (iv),hence we have to prove that the map mY : H0(N ′

L) → H0(L,NY |L) is onto. Thismap depends upon our complete intersection Y . We express it in terms of thesystem of equations b := (b1, · · · , bs) ∈ H0(IL(d)) of Y , rather than in terms of Yitself. This allows us to describe the associated morphism mY : N ′

L → NY |L or, viathe identifications of Lemma 4.4, mb : OL(1)

p ⊕OH(1) → OL(d) as follows:- for the first factor, the j-th component (1 ≤ j ≤ p), from OL(1) to OL(d), is

the derivative of b with respect to xj ;-for the second factor, from OH(1) to OL(d), we have the derivative of b with

respect to x0 (note that indeed this derivative factors through OH(1)).What we have to prove is that, for b sufficiently general, H0(mb) is onto. For

this, thanks to our surjectivity assumption, it is enough to prove that b → mb isdominant (or onto).

We prove that b → mb is onto. For this we take m := (m1, · · · ,mp,m0) :OL(1)

p⊕OH(1) → OL(d) and search for b with m = mb. By the standard cohomo-logical argument, we may lift m1, · · · ,mp and consider we are given (m1, · · · ,mp) :OPn(1)p → OPn(d). Now for m0, we see it as a section of Hom(OH ,OL(d − 1)),hence as a section of OL(d − 1) annihilated by x0, thus of the form txt−1

0 f withf a section of OL(d − t), using the characteristic zero assumption. As above, we

63


may lift f as a section, still denoted by f , of OPn(d− t). At this point we may setb := x1m1 + · · ·+ xpmp + xt

0f and check that it has the desired property.(ii) The proof is almost the same: we apply Lemma 4.5 (iv). This time, we

have to prove that the map mY : H0(N ′L) → H0(L,NY |L) sends K0(N ′

L) onto

K0(L,NY |L). As above we introduce a system of equations b := (b1, · · · , bs) ∈H0(IL(d)) of Y . Via the identifications of Lemma 4.4, we are concerned, forb sufficiently general, with the sheaf morphism mb : OL(1)

p ⊕ OH(1) → OL(d)defined by the same formulas as in the previous case.

Since our identifications send K0(N ′L) to K0(OL(1))

p ⊕ H0(OH(1)) andK0(L,NY |L) to K0(OL(d)), it is enough to prove that the image of b → mb is

K0(OL(1)p ⊕OH(1),OL(d)). For this we take

m := (m1, · · · ,mp,m0) ∈ K0(OL(1)p ⊕OH(1),OL(d)).

This means that m1, · · · ,mp come from sections still denoted by m1, · · · ,mp in

H0(Pn,O(d−1)), while m0 is of the form xt−10 f , or better of the form txt−1

0 f , withf a section of H0(Pn,O(d− t)). We now search for b with m = mb. Again we mayset b := x1m1 + · · ·+ xpmp + xt

0f and check that it has the desired property.�

5. Nonemptiness

In this section, we prove our conjecture 4.2 in the case we need. We restrictto the very special case where t is the greatest number in our sequence d, and weassume furthermore that t is at least 3, and that it occurs only once in d. We willprove:

Proposition 5.1. Under the above restrictions, when ρ is nonnegative, for anycomplete intersection Y of type d in Pn, the Hilbert scheme of t-fat r-planes in Yis nonempty.

We keep the notations of the previous section. Furthermore, we denote byh0(u, e) the number of monomials of degree e in u variables, and accordingly, for anysequence e := (e1, · · · , es) of integers, we set h0(u, e) := h0(u, e1) + · · ·+ h0(u, es).

Thanks to Proposition 4.6, it is enough to prove a maximal rank statement,which depends on whether r is zero or not. Namely, we have to prove the followingtwo lemmas.

Lemma 5.2. (the case r ≥ 1) For p satisfying (r+2)p+r+1 ≥ h0(r+2,d)−1,and for the general morphism m : OL(1)

p ⊕OH(1) → OL(d), H0(m) is onto.

Lemma 5.3. (the case r = 0) We suppose 2p + 1 ≥ h0(2,d) − 1. Then, forthe general morphism m in K0(OL(1)

p ⊕ OH(1),OL(d)), the image by H0(m) ofK0(OL(1)

p)⊕H0(OH(1)) in H0(OL(d)) is K0(OL(d)).

Recall that K0(OL(1)p⊕OH(1),OL(d)) denotes the image of the natural map

from K0(OL(d− 1))⊕K0(OL(d− t)) to Hom(OL(1)p ⊕OH(1),OL(d)).

The differences between our two lemmas can be erased by switching to thepoint of view of graded modules. So, just for the present section, we radicallychange the meaning of our notations: from now on, OL denotes the graded ringk[x0, · · · , xr+1]/(x

t0) and OH denotes the quotient graded module k[x1, · · · , xr+1].

For a graded OL-module G with graduation γ, we write G(a) for the module Gequipped with the graduation γa := γ − a. For a graded module G, by H0(G) we



mean the degree 0 component of G, while for a morphism m of graded modules,by H0(m) we mean the restriction of m to the degree 0 components. With theseconventions, our two lemmas rephrase as the single following one:

Lemma 5.4. For p satisfying (r + 2)p + r + 1 ≥ h0(r + 2,d) − 1, and for thegeneral morphism m : OL(1)

p ⊕OH(1) → OL(d), H0(m) is onto.

Here we use a method which can be tracked back to [EH, EHM], wheresimilar results were obtained in a different context. We denote by M the vectorspace Hom(OL(1)

p ⊕OH(1),OL(d)), by S the space of nontrivial linear forms onH0(OL(d)), and by Z the “incidence” subscheme in M×S consisting of pairs (m, )for which ◦H0(m) vanishes. We denote by the second projection: : Z → S.What we want to prove is that the first projection Z → M is not dominant.This will follow if we prove the inequality dimZ ≤ dimM , since the fibers ofour projection are unions of lines. We proceed by contradiction and suppose thatthe projection Z → M is dominant. To each λ ∈ S we attach the bilinear form λ∗

on H0(OL(d − 1)) ×H0(OL(1)) defined by λ∗(f, v) = λ(vf). By semi-continuity,we have an open subset Zu ⊂ Z which still dominates M , and where the rank u of∗ is constant.

In the first factor H0(OL(d− 1)) of our product, we have a distinguished line:the line D generated (in the summand OL(t)) by xt−1

0 . Our first observation is thefollowing:

Lemma 5.5. For our general point z ∈ Zu, ∗(z) vanishes on D ×H0(OL(1)).

Proof. In OL(d) we have the summand OL(t). And therein, we have thegraded submodule xt−1

0 OL(1) consisting of multiples of xt−10 . This submodule may

be better denoted by xt−10 OH(1) since the multiplication by xt−1

0 , which sendsOL(1) into OL(t), factors through OH(1). This submodule xt−1

0 OH(1) is easilyidentified as the submodule of OL(d) which is annihilated by x0. Hence, anymorphism m ∈ M has to send the second summand OH(1) of its domain, which isannihilated by x0, into the summand OL(t) of its codomain, and more precisely intothe submodule xt−1

0 OH(1) mentioned above. Also a sufficiently general morphismm ∈ M sends OH(1) isomorphically onto that submodule. Accordingly, H0(m)sends H0(OH(1)) isomorphically onto H0(xt−1

0 OH(1)). So, for our general z ∈Zu, (z) has to vanish on H0(xt−1

0 OH(1)) which implies that ∗(z) vanishes onD ×H0(OL(1)).

�

Our next observation stresses the role of u, which is to control the dimensionof the fiber of Z → S. We denote by Su the projection of Zu in S.

Lemma 5.6. The codimension of the fiber of Zu over a point λ ∈ Su is pu.

Proof. Let m := (m1, · · · ,mp;m0) be a point in M , where (m1, · · · ,mp) arein H0(OL(d− 1)) while m0 is in H0(OL(d− t)).

Thanks to the previous lemma, we see that λ ◦ H0(m) vanishes if and onlyif λ∗(m1) = · · · = λ∗(mp) = 0. Each one among these p equations imposes uindependent conditions on m, since the rank of λ∗ is u. Since these equationsconcern different components ofm, their ranks add up to the rank ofm → λ◦H0(m)which turns out to be pu.

�

65


Our next task consists in estimating the dimension of Su.

Lemma 5.7. The dimension of Su is at most h0(u,d) + (r + 1− u)u.

Proof. For this we have to single out the line E generated by x0 in H0(OL(1))and to distinguish two cases according to whether, for our general z ∈ Zu,

∗(z)vanishes or not on H0(OL(d− 1))× E.

(i) We start with the (slightly simpler) case where ∗(z) does not vanish onH0(OL(d− 1))× E.

In order to bound the dimension of Su at a point λ0, we will define, in aneighborhood U ⊂ Su of λ0, two algebraic maps f : U → Ab and g : U → Ac

so that (f, g) is injective. This will bound the dimension of Su by b + c. To thiseffect, we reorder our basis C := (x0, · · · , xr+1) of H0(OL′(1)) (where x0 remainsan equation of H) so that, in this basis, the first u rows of the matrix of λ∗

0 arelinearly independent. This property will hold in a neighborhood of λ0 which wetake as U . We write C ′ for the sub-basis (x0, · · · , xu−1) and C ′′ for the rest of thebasis so that we have C = C ′�C ′′. Next, in OL(d), we have the basis consisting ofmonomials in each summand, which we call d-monomials. Similarly, we have thebasis of (d− 1)-monomials in OL(d− 1).

Associated with these bases, we have the matrix Nλ of λ∗, which is an algebraicfunction of λ. Now for each element in C ′′, we have the u coordinates of thecorresponding row (in Nλ) as a combination of the rows in C ′. This defines b :=(r + 2− u)u functions on U which altogether yield our map f .

Now for g(λ) we take the restriction λ′ of λ to the subspace generated by thefollowing set T ′ of d-monomials: first, take the set T of d-monomials depending(at most) on variables in C ′, then delete those, in the summand OL(t), which aredivisible by xt−1

0 . This deletion corresponds to the fact observed above that λvanishes there.

What we have to check is that λ is determined by λ′ and f(λ). For this, weclaim that for each integer q with 0 ≤ q ≤ t the values of λ on the set Tq ofthose d-monomials which are of degree q with respect to variables in C ′′ are linearcombinations (where coefficients are polynomials in f(λ)) of its values on T ′. Weprove the claim by induction on q, the case q = 0 following from the vanishingmentioned above. For the general case we consider a d-monomial m := m′xi wherem′ is a (d − 1)-monomial and xi is in C ′′. In the column corresponding to m′ inNλ, the first u entries are values of λ on elements of Tq−1, while the entry in therow corresponding to xi is λ(m), which gives us the desired linear relation.

It remains to check that the number of elements in T ′ is h0(u,d)−u. Indeed, 1is subtracted from h0(u,d) because, although d contains t, xt

0 is not a d-monomial,and u−1 is subtracted due to the difference between T and T ′. Thus the codomainof our map g is Ac with c := h0(u,d)− u.

(ii) Now we treat the similar case where ∗(z) vanishes on H0(OL(d−1))×E.The method is the same so we just highlight the changes. Thanks to the vanishingassumption, λ∗ is now determined by the bilinear form λ′∗ induced on H0(OL(d−1))×H0(OH(1)). Our basis C now has the form (x1, · · · , xr+1), and the subbasisC ′ is (x1, · · · , xu) Accordingly, the number b is now equal to (r + 1− u)u. On theother hand, here, there is no deletion, T ′ is equal to T and its number of elementsis h0(u,d), which yields the desired formula. �



In order to complete the proof of 5.2, it remains to check that the estimatesobtained so far make the dimension of Zu smaller than that of M , namely that thecodimension (in M) obtained for the fiber of Zu → Su is bigger than the dimensionof Su. This reads:

Lemma 5.8. For t ≥ 3, p satisfying (r + 2)p + r + 1 ≥ h0(r + 2,d) − 1, and1 ≤ u ≤ r + 2 we have h0(u,d) + (r + 1− u)u ≤ pu.

Proof. We argue by convexity (with respect to u) and start by checking theextreme cases:

(i) For u = r + 2, the desired conclusion is just the assumption.(ii) For u = 1, we contrapose and prove that p ≤ h0(1,d) + r − 1 implies

(r + 2)p+ r + 3 ≤ h0(r + 2,d). Taking the critical value s+ r − 1 for p we have toprove s(r + 2) + (r + 1)2 ≤ h0(r + 2,d).

We split this inequality summand by summand, in other words we claima) r + 2 + (r + 1)2 ≤ h0(r + 2, t) (for the occurence of t in d) andb) r + 2 ≤ h0(r + 2, δ) (for each other integer, δ ≥ 2, in d).For a) it is sufficient to check the first case t = 3. In this case, we have to prove

6(r+ 1)2 ≤ (r+ 2)[(r+ 3)(r+ 4)− 6] or, dividing by r+ 1, 6r+ 6 ≤ (r+ 2)(r+ 6),or 0 ≤ r2 + 2r + 6, which is evident. Moreover, b) is clear since for each variablexi, we have the monomial xδ

i .It remains to check that the function f := u → h0(u,d) + (r + 1 − u)u is

convex on our interval [1, r + 2]. For this, we compute the discrete derivativesf ′ := u → f(u+1)− f(u) and f ′′. We find f ′(u) = h0(u+1,d−1)+ r+1−2u−1and f ′′(u) = h0(u+2,d−2)− 2. We see that this second derivative is nonnegativefor u ≥ 1, yielding the desired convexity. �

6. Spannedness

This section is devoted to the proof of the desired covering statement :

Proposition 6.1. Let r ≥ 0, 1 ≤ s ≤ n− r − 1 and 2 ≤ d1 ≤ ... ≤ ds−1 < dsbe integers. We set ρ := (r+ 2)(n− r)−Σs

i=1

(di+r+1r+1

)and assume the (necessary)

inequality

ρ+ r ≥ n− s.

If Y ⊂ Y ′ is any pair of type (d1, ..., ds) in Pn, then Y is covered by strong r-planes.

Thanks to Proposition 2.3, this statement is an immediate consequence of thefollowing one.

Proposition 6.2. Under the same assumptions, Y is covered by ds-fat r-planes.

We pose t := ds.From §4, we have the restricted flag-Hilbert scheme D′ ⊂ H′

1 × H2. Overthe first factor H′

1, we have the universal t-fat r-plane, say L ⊂ H′1 × Pn. Over

the second factor H2, we have the universal complete intersection of type d, sayY ⊂ H2 × Pn, and over D′, we have the universal flag, say L ⊂ Y ⊂ D′ × Pn.We have a natural projection e : Y → Y , and what we have to prove is that therestriction e′ : L → Y is onto.

67


Since e′ is a H2-morphism between varieties which are projective over H2, itsimage is also projective over H2. So it is sufficient to prove that e′ is dominant, and,for that, to find one point in L where the fiber of e′ has the expected dimensionρ+ r − n+ s, and not more.

So we compute the fiber of e′ at a point (L, Y, p) ∈ L, where L is a t-fat r-planecontained in the complete intersection Y and p is a point on L. This splits into twocases according to whether r is zero or not.

(i) The case r = 0. This case is known since [Ro]. Hence we just give the ideaof the proof, which is similar to but simpler than the other case. The variety oft-fat points at p contained in Y is identified with a subvariety in the projectivizedtangent space of Y at p with equations depending on the equations of Y . Thenumber of these equations is easily checked to be Σi(di − 1) − 1: di − 1 is thenumber of degrees between 2 and di, and 1 is subtracted for the degree t. Thanksto our assumption on ρ, this is at most n − s − 1 which is the dimension of thisprojective space. Hence this variety is nonempty.

(ii) The case r ≥ 1. We consider the subscheme WY,p in the projectivized tan-gent space PTY,p of Y at the point p, which parametrizes lines through p containedin Y . It is defined by the homogeneous components of the Taylor expansions at pof the equations of Y . For this reason, we setd′ := (1, 2, · · · , d1, · · · , 1, 2, · · · , ds) and d′′ := (2, · · · , d1, · · · , 2, · · · , ds).

We immediately observe that there is a natural isomorphism between the vectorspace of tuples of equations of type d (in n+1 variables) vanishing at p and tuplesof equations of type d′ (in n variables). This yields:

For (Y, p) sufficiently general in Y , WY,p is a sufficiently general complete in-tersection of type d′′ in PTY,p.

Next we have the following

Lemma 6.3. For any (Y, p) ∈ Y, the fiber of e′ over (Y, p) is isomorphic withthe Hilbert scheme of t-fat (r − 1)-planes in WY,p.

First we check how we may complete the proof of Proposition 6.2 using thislemma. Thanks to Proposition 5.1, we just have to check that the expected di-mension ρ′ of the Hilbert scheme of t-fat (r − 1)-planes in the generic completeintersection of type d′ in Pn−1 or equivalently of type d′′ in Pn−s−1 is nonnegative.We have

ρ′ = (r + 1)(n− r)− 1− Σsi=1Σ

dij=1

(j + r

r

)

= (r + 1)(n− r)− 1− Σsi=1(

(di + r + 1

r + 1

)− 1)

= (r + 2)(n− r)− 1− n+ r + s− Σsi=1

(di + r + 1

r + 1

)

= ρ+ r − n+ s.

This is nonnegative by assumption.Now we prove Lemma 6.3. First of all, we have a natural isomorphism g between

the Hilbert scheme of t-fat r-planes in Pn passing through p and the Hilbert schemeof t-fat (r − 1)-planes in the projectivized tangent space PTPn

p : if we identify thisprojectivized tangent space with a hyperplane K ⊂ P

n not passing through p, g(L)is the scheme-theoretic intersection of L with K.



Now we prove that g induces a bijection from the Hilbert scheme H ′ of t-fatr-planes in Y passing through p to the Hilbert scheme H ′′ of t-fat (r − 1)-planesin WY,p. Let M be a t-fat (r − 1)-plane in WY,p and M ′ ⊂ P

n be the unique t-fatr-plane through p corresponding to M (hence g(M ′) = M). Let (fi := Σj≤di

fij)i≤s

be a system of equations of Y . Note that the fi’s vanish at p. Our claim is that thefi’s all vanish on M ′ if and only if the fij all vanish on M = g(M ′). This followsreadily from the particular case of a single equation f , with n = r + 1, which westate explicitly:

Let f(x0, x′) := x0f1(x

′) + · · · + xδ0fδ(x

′) be a homogeneous polynomial inn + 1 variables x0, · · · , xn where x′ stands for (x1, · · · , xn). Note that f vanishesat p := (1, 0, · · · , 0). We denote by K the hyperplane defined by x0 = 0, by M thet-fat (n− 1)-plane defined by xt

1 = 0 and by N the t-fat (n− 2)-plane defined in Kby the same equation. Since f is a multiple of xt

1 if and only if all the fi’s are, wehave that f vanishes on M if and only if the fi’s vanish on N .

�

References

[Bl] S. Bloch, Lectures on Algebraic Cycles. Duke Univ. Math. Series IV, Durham, N.C.,1980.

[De-Ma] O. Debarre, L. Manivel, Sur la variete des espaces lineaires contenus dans une intersec-tion complete. (French) [The variety of linear spaces contained in a complete intersection]Math. Ann. 312 (1998), no. 3, 549–574.

[De] P. Deligne, Groupes de monodromie en geometrie algebrique. II. (French) Seminaire deGeometrie Algebrique du Bois-Marie 1967–1969 (SGA 7 II). Dirige par P. Deligne et N.Katz. Lecture Notes in Mathematics, Vol. 340. Springer-Verlag, Berlin-New York, 1973.x+438 pp.

[De-Di] P. Deligne, A. Dimca, Filtrations de Hodge et par l’ordre du pole pour les hypersurfacessingulieres. (French) [Hodge filtration and filtration by the order of the pole for singularhypersurfaces] Ann. Sci. Ecole Norm. Sup. (4) 23 (1990), no. 4, 645–656.

[EH] Ph. Ellia, A. Hirschowitz, Voie ouest I: Generation de certains fibres sur les espacesprojectifs et application, J. of Alg. Geom. 1 (1992) 531–547.

[EHM] Ph. Ellia, A. Hirschowitz, L. Manivel, Le probleme de Brill-Noether pour les fibres deSteiner et application aux courbes gauches. Annales ENS Paris 32 (1999) 835–857.

[Es] H. Esnault, Hodge type of subvarieties of Pn of small degrees. Math. Ann. 288 (1990),no. 3, 549–551.

[EsNS] H. Esnault, M.V. Nori, V. Srinivas, Hodge type of projective varieties of low degree.Math. Ann. 293 (1992), no. 1, 1–6.

[EsLV] H. Esnault, M. Levine, E. Viehweg, Chow groups of projective varieties of very smalldegree, Duke Math. J. 87, (1997), no. 1, 29–58.

[Fu] W. Fulton, Intersection theory, Ergeb.der Math.und ihrer Grenz. 3. Folge, Springer-Verlag, Berlin, 1984.

[Kl] J.O. Kleppe, Nonreduced components of the Hilbert scheme of smooth space curves.Space curves (Rocca di Papa, 1985), 181–207, Lecture Notes in Math., 1266, Springer,Berlin, 1987.

[Le] J.D. Lewis, Cylinder homomorphisms and Chow groups. Math. Nachr. 160 (1993), 205–221.

[Ot] A. Otwinowska, Remarques sur les groupes de Chow des hypersurfaces de petit degre, C.R. Acad. Sci. Paris Ser. I Math. 329 (1999), no. 1, 51–56.

[Pa] K. Paranjape, Cohomological and cycle-theoretic connectivity, Ann. of Math. (2) 139(1994), 641–660.

[Ro] A. Roitman, Rational equivalence of zero cycles, Mat. Zametki 28 (1980), no. 1, 85–90,169.

[Sc] C. Schoen, On Hodge structures and nonrepresentability of Chow groups, CompositioMath. 88 (1993), no. 3, 285–316.

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[Se] E. Sernesi, Deformations of algebraic schemes, Grundlehren der mathematischen Wis-senschaften, Springer, 2006, xi+339 pp.

CNRS, Laboratoire J.-A.Dieudonne, Universite de Nice–Sophia Antipolis, Parc Val-

rose, 06108 Nice Cedex 02, France


The Institute of Mathematical Sciences, CIT Campus, Taramani, Chennai 600113,

India

Current address: Department of Mathematics and Statistics, University of Hyderabad, Gachi-bowli, Central University P O, Hyderabad-500046, India




Vector Bundles and the Icosahedron

Nigel Hitchin

Dedicated to S. Ramanan on the occasion of his 70th birthday

Abstract. A plane curve C defined by a homogeneous polynomial satisfyingLaplace’s equation appears canonically as the vanishing of the Pfaffian of askew-symmetric matrix of linear forms. As a consequence there is a naturalsemi-stable rank two vector bundle defined on C. We consider the case ofdegree 3, and apply Atiyah’s classification of bundles to determine variousinvariant strata in the space of harmonic cubics. We encounter the Mukai-Umemura threefold and link up with the classical geometry of the Clebschdiagonal cubic surface, certain distinguished rational curves on it, and theaction of the icosahedral group.

1. Introduction

Describing the equation of a plane curve C as the determinant of a matrixof linear forms is a classical problem. A related issue is expressing C as thePfaffian of a skew-symmetric matrix of forms. When the curve is defined by aharmonic homogeneous polynomial (a solution to Laplace’s equation), then the ac-tion of the Lie algebra of the orthogonal group SO(3,C) provides a natural suchexpression, and with it, a natural semi-stable rank two vector bundle E on C.This paper concerns itself with these bundles, which have trivial determinant andH0(C,E) = 0.

For a cubic curve, Atiyah’s classification shows that they fall into three types:a sum of line bundles L ⊕ L∗ with L2 nontrivial, a non-trivial extension of a linebundle of order two by itself, and a trivial extension. We shall show that the secondand third cases are described in the space of harmonic cubics by the vanishing of acertain invariant polynomial of degree six.

Our route to this result uses the Mukai-Umemura threefold and we link up withthe classical geometry of the Clebsch diagonal cubic surface, certain distinguishedrational curves on it, and the action of the symmetric group S5.

The original motivation for this paper comes from the author’s interest in ex-plicit solutions to Painleve equations, as in [10] and [11]. These were connected tothe study of certain threefolds with an open orbit of SO(3,C) and finite stabilizer.

2010 Mathematics Subject Classification. Primary 14M12; Secondary 14H60.


1



71

2 NIGEL HITCHIN

The Mukai-Umemura manifold has such an action where the stabilizer is the icosa-hedral group. Since the first paper, where the stabilizer is the dihedral group, waswritten for the 60th birthdays of Narasimhan and Seshadri, it seems appropriate,sixteen years on, to discuss the icosahedron for Ramanan.

2. Representations of SO(3,C)

Let V be a finite-dimensional irreducible representation space of the groupSO(3,C). These occur in each odd dimension (2d+ 1) and have an invariant innerproduct (u, v). Restricted to the circle subgroup SO(2,C) ⊂ SO(3,C), the weightsare −d ≤ n ≤ d; in particular the weight zero occurs with multiplicity one so thereis a unique invariant element. The usual realization of V is as S2d – the space ofhomogeneous polynomials in the complex variables z1, z2 of degree 2d under theaction of SL(2,C)/± 1 ∼= SO(3,C).

The Lie algebra so(3) acts on V as skew-adjoint transformations and so eachx ∈ so(3) defines a skew form

ωx(u, v) = (x · u, v).Now fix v ∈ V and restrict ωx to the 2d-dimensional orthogonal complement W ofv. The Pfaffian ωx ∧ ωx · · · ∧ ωx ∈ Λ2dW ∗ ∼= C defines a homogeneous polynomialf(x) of degree d.

The fact that we have a natural map from vectors in V to polynomials is nosurprise if we recall that another realization of the (2d+1)-dimensional irreduciblerepresentation of SO(3,C) is as spherical harmonics – homogeneous polynomialsf(x1, x2, x3) of degree d which satisfy Laplace’s equation. However, v gives some-what more than just a polynomial.

The skew form ωx defines a map ωx : W → W ∗, which is linear in x. Hence ifP2 is the projective space of the vector space so(3), we have a sequence of sheaves

(2.1) 0 → OP2(W (−2))ω→ OP2(W ∗(−1)) → E → 0

where E is a rank 2 vector bundle supported on the curve C ⊂ P2 defined byf(x) = 0. The vector bundle satisfies Λ2E ∼= KC , the canonical bundle, and fromthe exact cohomology sequence, H0(C,E) = 0.

These facts about curves defined by Pfaffians (and much more besides) can befound in [3]. In particular, the bundle is always semi-stable. To see this, note thatif L ⊂ E is a subbundle, the inclusion defines a section i of L∗E and if degL ≥ gthen by Riemann-Roch there is a non-zero section s of L. But then si is a nonzerosection of E which is a contradiction. Hence we must have

degL ≤ g − 1 =1

2degKC =

1

2degE.

This natural process thus generates a curve and a rank 2 semi-stable bundle,and the question we ask is “What is this bundle?”

When d = 3, the curve C is a plane cubic and then we know from Atiyah’sclassification of bundles on an elliptic curve [1] that, when C is smooth, there areonly three possibilities. Since KC

∼= O, Λ2E is trivial and H0(C,E) = 0 and wehave the cases:

• E = L⊕ L∗ where degL = 0 and L = O (the generic case)

72

VECTOR BUNDLES AND THE ICOSAHEDRON 3

• E is a non-trivial extension 0 → L → E → L → 0 where L2 ∼= O andL = O

• E = L⊕ L where L2 ∼= O and L = O.

These three types define a stratification of the six-dimensional projective spaceP(V ) where V is the seven-dimensional space of harmonic cubic polynomials f – orat least that part of it for which the curve f(x) = 0 is nonsingular. To investigatethis further we have to see a link with another piece of geometry.

3. Isotropic spaces

Let d = 3, and as above let f ∈ V define a smooth cubic curve C ⊂ P2 andW ⊂ V the six-dimensional orthogonal complement of f in W . Since Λ2E ∼= O, wehave E∗ ∼= E and then the kernel of ω : W → W ∗(1) on C can be identified withE(−1).

Suppose U ⊂ W is a three-dimensional subspace, isotropic with respect to theskew forms ωx for all x. Then, for each x, ωx : W → W ∗ maps U to its annihilatorUo and so defines a homomorphism of sheaves on P2, ω|U : U → Uo(1). Moreoverthe determinant of this map is the Pfaffian of ωx. The kernel of ω|U on C is thena line bundle L(−1) ⊂ U . Since this is the restriction of ω to U , L is a subbundleof E. From the treatment of determinantal loci in [3] we have degL = 0 andH0(C,L) = 0 so L = O.

Conversely, on C, suppose we have a line bundle L ⊂ E with degL = 0 andL = O. From the exact cohomology sequence of (2.1) we have W ∗ ∼= H0(C,E(1)).If α ∈ W ∗ annihilates L(−1) then it defines a section of (E/L)(1) = L∗(1) which isof degree 3 and so has a three-dimensional space of sections on C. It follows thatL(−1) ⊂ W sweeps out a three-dimensional subspace U .

Now ω|U is a section of Λ2U∗(1) on C and we have Λ2U∗ ∼= U ⊗ Λ3U∗ so thisdefines a section w of U(1) which tautologically is annihilated by ω. Thus if w = 0,then kerω|U ∼= O(−1). But kerω|U ∼= L(−1) and L = O, so we deduce that w = 0and U is isotropic for each ωx.

If we look again at the trichotomy of Atiyah’s classification, we see that thethree cases are equivalent to

• E has two rank one subbundles• E has one rank one subbundle• E has infinitely many rank one subbundles

and by what we have just seen, this condition translates into an equivalent state-ment about the three-dimensional isotropic subspaces of W :

• W has two isotropic subspaces• W has one isotropic subspace• W has infinitely many isotropic subspaces.

This provides an extension of the criterion to any f ∈ V , and not just thosewhich define smooth cubic curves. (In fact, Atiyah’s classification has been extendedin [5] to a class of singular elliptic curves; moreover the wild case of cuspidal cubicsdoes not occur when f is harmonic).

4. The Mukai-Umemura threefold

The study of three-dimensional isotropic subspaces of V is best approachedvia a special Fano threefold introduced by Mukai and Umemura [15]. We consider

73

4 NIGEL HITCHIN

the Grassmannian G(3, V ) of three-dimensional subspaces of the seven-dimensionalrepresentation space V and its universal rank 3 bundle E . For x ∈ so(3) we havethe skew form ωx on V , and so a section of the rank 9 vector bundle over G(3, V ):

Λ2E∗ ⊗ so(3).

Now dimG(3, V ) = 3× (7−3) = 12 and then the zero set of the section is a smooth12 − 9 = 3-dimensional manifold known as the Mukai-Umemura threefold Z. Byconstruction it parametrizes subspaces U isotropic for all ωx and thereby has anatural action of SO(3,C).

The threefold Z has the same additive integral cohomology as P3 and H2(Z,Z)is generated by x = c1. However x2 ∈ H4(Z,Z) is 22 times a generator y. Sincec1 > 0 the Todd genus is 1 so c1c2 = 24 and we have Chern classes

(4.1) c1 = x c2 = 24y c3 = 4xy.

Given f ∈ V , the inner product (f,−) defines an element of V ∗ and, restrictingto the universal bundle E ⊂ V , we get a section of E∗. Over Z, this is a rankthree bundle and the section vanishes at the points which correspond to isotropicthree-dimensional subspaces orthogonal to f .

Proposition 4.1. On the Mukai-Umemura threefold, c3(E∗) = 2.

Proof. The tangent bundle of the Grassmannian is Hom(E , V/E) and Z is thenon-degenerate zero set of a section of Λ2E∗ ⊗ C

3. Thus, as C∞ bundles

TZ ⊕ (Λ2E∗ ⊗ C3) ∼= Hom(E , V/E).

Applying the Chern character we find

c1(E∗) = c1 c2(E∗) = c21 −1

2c2 c3(E∗) =

1

10(c3 + 4c31 − 3c1c2).

From (4.1) we obtain 10c3(E∗) = 4 + 88− 72 = 20 and hence the result. �

A section with nondegenerate zero set will thus vanish at two points. We cannow see the trichotomy in terms of the section s of E∗:

• s vanishes at two points• s vanishes at one point• s vanishes on a subvariety of positive dimension.

5. The icosahedron

The ten-dimensional space of all homogenous cubics in x has an SO(3,C)-invariant product which can be normalized so that (f(x), (x, a)3) = f(a). For fixeda ∈ C3, the polynomial (x, a)3 is not harmonic but its orthogonal projection ontoV is

fa(x) = (x, a)3 − 3

5(a, a)(x, a)(x, x).

This is invariant by the group of rotations fixing a. For any cubic f we still havethe inner product property (f, fa) = f(a).

The action of u ∈ so(3) on fa is

(u · fa)(x) = 3(x, [u, a])[(x, a)2 − 1

5(a, a)(x, x)].

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Take a regular icosahedron with vertices at ±a1,±a2 . . . ,±a6. The angle θ be-tween any two axes joining opposite vertices satisfies cos θ = ±1/

√5. Now consider

(u · fai, faj

) = 3(aj , [u, ai])[(aj , ai)2 − 1

5(ai, ai)(aj , aj)].

If i = j then (aj , [u, ai]) = 0 and when i = j

(aj , ai)2 − 1

5(ai, ai)(aj , aj) = (cos2 θ − 1

5)(ai, ai)(aj , aj) = 0

so fa1, . . . , fa6

span an isotropic subspace U , invariant under the icosahedral groupG. Its orthogonal complement in V consists of those f such that

0 = (f, fai) = f(ai) = 0.

There are five objects which are permuted by G which realize the well-knownisomorphism G ∼= A5. These are the five sets of three orthogonal planes in whichall 12 vertices lie. Let e1, e2, e3 be the three unit normals to such a set thenthe cubic (e1, x)(e2, x)(e3, x) vanishes at ai as do all its transforms by G. Thesespan the four-dimensional permutation representation 4 of A5. Moreover, since(e1, x)(e2, x)(e3, x) satisfies Laplace’s equation, this is the orthogonal complementof U in V , hence U has dimension 3. Every icosahedron therefore defines a pointin the Mukai-Umemura manifold and the three-dimensional orbit SO(3,C)/G is adense open set in Z.

Generically, a harmonic cubic f defines a section s of E∗ which vanishes at twopoints in this open set, which means that the corresponding plane cubic C containsthe twelve points in P2 defined by the axes of two icosahedra. Put another way, ifa generic cubic curve contains one such “icosahedral set” {[a1], [a2], . . . , [a6]} ⊂ P2

then it contains another.

6. Degenerate icosahedra

6.1. The two types. We need to understand also the divisor of “degenerateicosahedral sets” which forms the complement of the open orbit of SO(3,C) in Z.These correspond to isotropic subspaces of two types, constituting two orbits, ofdimension 2 and 1 respectively.

A representative of the first type is the space with basis

(b, x)3 − 3

5(b, x)(b, b)(x, x), (a, x)2(b, x), (a, x)3

where (a, a) = 0, (a, b) = 0 and (b, b) = 0. This subspace is invariant by therotations about the axis b and is spanned by the weight spaces {0, 2, 3} for thataction. Geometrically, the isotropic subspace is defined by [b] ∈ P2 (a point noton the null conic Q defined by (x, x) = 0) together with the point of contact [a] ofa tangent to Q through [b]. The choice of tangent is a double covering of P2 \ Q,which is abstractly an affine quadric.

The icosahedral set here describes five of the vectors {a1, a2, . . . , a6} coalescinginto a single vector a. Under this degeneration the relation (aj , ai)

2− 15 (ai, ai)(aj , aj)

= 0 implies that a is null and the sixth vector b satisfies (b, a) = 0.The second type is a subspace with basis

(a, x)((b, x)2 − 1

5(b, b)(x, x)), (a, x)2(b, x), (a, x)3

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6 NIGEL HITCHIN

where (a, a) = 0 and (a, b) = 0. This is geometrically defined by the point [a] ∈ Qand is invariant by the Borel subgroup which fixes [a]. Decomposing with respectto a semisimple element this is the span of weight spaces {1, 2, 3}. Here all sixvectors {a1, a2, . . . , a6} coalesce into a single null vector a.

The union of these two orbits forms an anticanonical divisor in Z, for if X1,X2, X3 are the vector fields on Z generated by a basis of so(3), then X1 ∧X2 ∧X3

vanishes on the lower-dimensional orbits and this is a section of K∗Z = Λ3TZ. Since

c1(Z) is a generator of H2(Z,Z) it must vanish with multiplicity 1. Note that thisanticanonical divisor D cannot be smooth (for then it would be a K3 surface ora torus). It is instead a singular image of a map α : P1 × P1 → Z (in fact, atransverse to the diagonal in P1 ×P1 has a cusp singularity y2 = x3 (see [7]).

To define α recall that we have identified the type 1 orbit as an affine quadric,the complement of the diagonal in P1×P1. This is an ordered pair of points [a], [a′]on the conic Q ∼= P1, where the two tangents meet at [b] ∈ P2. To extend the mapα to the diagonal put c = a+ tb, with (c, a) = 0 and let t → 0. Then

(c, x)3 − 3

5(c, x)(c, c)(x, x) =

= (a, x)3 + 3t(a, x)2(b, x) + 3t2[(a, x)((b, x)2 − 1

5(b, b)(x, x))] + . . .

and the three leading coefficients span the type 2 isotropic subspace.Consider a degenerate icosahedral set of type 1, spanned by the three cubic

polynomials (b, x)3−3(b, x)(b, b)(x, x)/5, (a, x)2(b, x) and (a, x)3. Then f is orthog-onal to this if the curve C given by f(x) = 0 intersects the conic Q tangentially at[a] and passes through [b]. For type 2, the intersection multiplicity of C with Q at[a] must be at least 3.

6.2. The universal bundle. We should also consider the universal bundle Eon the divisor D, or rather its pullback α∗E on P1 ×P1. Note that both types ofdegenerate isotropic subspace contain (a, x)3 and (a, x)2(b, x) where (b, a) = 0, orequivalently the two-dimensional subspace given by (a, x)2(c, x) for all c orthogonalto a.

We use the two-fold covering map π : P1 × P1 → P2, the quotient by theinvolution interchanging the factors. Each factor is isomorphic to the diagonalwhich maps to the conic Q. As usual, geometrically π([a], [a′]) is the point [b] ofintersection of the tangents at [a], [a′] ∈ Q.

The two-dimensional vector space a⊥ ⊂ C3 for [a] ∈ Q defines a vector bundleA over Q for which the projective bundle P(A) is trivial. Indeed, it is the bundle oftangent lines to the conic and so under the map π can be identified with the familyP1 × {x}, x ∈ P1, which is a trivial bundle over the diagonal. On the other handsince (a, a) = 0

a⊥ ∼= (C3/OP2(−1))∗

and so Λ2A ∼= OP2(−1) = O(−2), identifying Q with P1. Hence, since P(A) istrivial,

A ∼= O(−1)⊗ C2.

Multiplying by the factor (a, x)2 which is quadratic in a, it follows that the sub-bundle in α∗E of cubics of the form (a, x)2(c, x) with (c, a) = 0 is isomorphic toO(−5, 0)⊗ C2.

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The cubic (b, x)3 − 3(b, x)(b, b)(x, x)/5, together with the subspace A, spans atype 1 degenerate subspace for (b, b) = 0 and when (b, b) = 0 it lies in A. This termis homogeneous of degree 3 in b, and so defines a homomorphism from π∗OP2

(−3) =O(−3,−3) to α∗E . It projects to a section of the line bundle α∗E/A which vanisheson the diagonal Δ ⊂ P1 ×P1. Hence α∗E/A is of the form O(k, k) and

Λ3α∗E ∼= O(k − 10, k).

But for Z we have the Chern number c31 = 22 and since D is an anticanonicaldivisor, c21[D] = 22. However c1(E) = −c1(Z) and so

22 = 2k(k − 10)

and k = −1. Thus α∗E is an extension

0 → O(−5, 0)⊗ C2 → α∗E → O(−1,−1) → 0.

This extension is classified by an element of H1(P1 × P1,O(−4, 1)) ⊗ C2. It isalso by definition SO(3,C)-invariant by the diagonal action on P1 × P1. But asrepresentation spaces

H1(P1 ×P1,O(−4, 1))⊗C2 ∼= H1(P1,O(−4))⊗H0(P1,O(1))⊗C

2 ∼= S2 ⊗ S ⊗ S

and α∗E is defined by a non-zero vector in the unique invariant one-dimensionalsubspace.

The sections of α∗E∗ fit into the exact sequence

0 → H0(P1 ×P1,O(1, 1)) → H0(P1 ×P1, α∗E∗) → H0(P1,O(5))⊗ C2 → 0.

As a representation space we then have

H0(P1 ×P1, α∗E∗) ∼= S ⊗ S + S ⊗ S5

which contains with multiplicity one the seven-dimensional representation S6 as asubspace of S ⊗ S5. These sections are the restriction to D of the sections (f,−)on Z that we have considered earlier. There is one important consequence of this:

Proposition 6.1. A harmonic cubic f is orthogonal to a degenerate isotropicsubspace if and only if Δ(f) = 0 for a certain SO(3,C)-invariant polynomial Δ ofdegree 10.

Proof. We want to know when a section s of E∗ defined by (f,−) vanisheson D. If it does vanish somewhere then the map to H0(P1,O(5)) ⊗ C2 gives twosections of O(5) with a common zero. These arise from S6 ⊂ S5⊗S so the conditionis that we have a homogeneous polynomial p(z1, z2) of degree 6 such that the twopartial derivatives ∂p/∂z1, ∂p/∂z2, homogeneous of degree 5, have a common zero.The vanishing of the resultant is the condition. This is a degree 10 polynomial Δin the coefficients of f , the discriminant. Its vanishing implies that there is a point[a] = [z1, z2] ∈ P1 where the section of O(6) has a double zero. Then the sectionof O(5, 0)⊗ C2 vanishes on {[a]} ×P1 ⊂ P1 ×P1.

The dual of the inclusion O(−3,−3) ⊂ α∗E gives a homomorphism

α∗E∗ → O(3, 3)

which determines the third component of the section s. It maps H0(P1 × P1, E∗)to

H0(P1 ×P1,O(3, 3)) ∼= S3 ⊗ S3

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8 NIGEL HITCHIN

and our seven-dimensional representation space S6 maps into the symmetric ele-ments. Such a section is then the pull-back of a section of OP2(3) - the cubic f . Inparticular, its divisor passes through ([a], [a]) where the other two components of svanish.

We see then that the section s vanishes at a point of D if the discriminantvanishes. Geometrically this implies that the cubic curve C meets the null conictangentially at [a]. �

7. The Clebsch cubic surface

The standard permutation representation of A5 is the four-dimensional sub-space 4 ⊂ C5 defined by y1 + y2 + · · · + y5 = 0. We shall investigate next thegeometry of the corresponding three-dimensional projective space P3.

The ring of invariants for A5 on 4 is generated by the elementary symmetricfunctions σ2, σ3, σ4, σ5 in (y1, y2, . . . , y5) and the degree 10 invariant∏

i<j

(yi − yj).

The invariant y21 +y22 + · · ·+y25 defines a nonsingular quadratic form on 4 and thenorthogonality of subspaces defines polarity in P3: each point has a polar plane, andeach line a polar line. There is a unique invariant cubic, which we can write aseither σ3 = 0 or more usually as

y31 + y32 + · · ·+ y35 = 0.

This equation defines the Clebsch cubic surface S. It is nonsingular and is invariantunder the action of the full symmetric group S5.

Consider now six points [a1], [a2], . . . , [a6] ∈ P2 forming an icosahedral set.No three are collinear and no conic passes through them all, for then it would beinvariant by the icosahedral group and so given by the null conic (x, x) = 0; butthe vertices ai are not null. It follows that blowing up the six points gives a non-singular cubic surface in P3. More precisely, the embedding is given by the planecubic curves that pass through the six points, which is the representation 4 of A5.Hence, by uniqueness, the invariant cubic surface must be the Clebsch surface S.

The blown up points form six disjoint lines in S, half of a double-six configu-ration. The other six are given by the proper transforms of conics passing throughfive of the six points. These we have encountered already:

(ai, x)2 − 1

5(ai, ai)(x, x) = 0.

Blowing these down gives another map from S to a projective plane P2. More-over since the icosahedral group permutes these lines, it is the plane of a three-dimensional representation. These two planes are the projective spaces of the twoinequivalent three-dimensional representations 3 and 3 of A5. (These can be viewedas the self-dual and anti-self-dual two-forms on 4).

Since the null conic Q ⊂ P2 does not meet the points [ai] to be blown up, it liftsto a rational curve R ⊂ S with self-intersection number 4, and hence of degree 6 inP3. The null conic in the second projective plane similarly lifts to a degree 6 curveR ⊂ S. Each of these is individually invariant by the group A5 and interchangedby S5.

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The rational curves R and R have a rather special relationship: the polar planeof a point p ∈ R is a tritangent plane to R and vice-versa. Furthermore the polarline of the tangent line through p is a trisecant of R. This classical result can befound in [14]. It establishes a 3 : 3 correspondence between the two curves.

8. The trichotomy for P3

So far we have introduced three pieces of geometry – the vector bundle ona harmonic cubic curve, the Mukai-Umemura threefold, and the projective spaceP3 of the permutation representation of A5. We seek the distinguished SO(3,C)-invariant subvarieties in the six-dimensional projective space P(V ) which describethe three different types of bundles. To do this we first restrict to P3 ⊂ P(V ),where we shall see the classical geometry described in the previous section playinga role. The first step is to establish a rational correspondence between the Mukai-Umemura threefold and P3.

8.1. From Mukai-Umemura to P3. The Mukai-Umemura threefold param-etrizes three-dimensional isotropic subspaces of V . Fix a nondegenerate one, U0,defined by the icosahedral set [a1], [a2], . . . , [a6] as a base point. For any other Uwe can consider the subspace U +U0 ⊂ V . Generically this has dimension 6 and sohas a one-dimensional orthogonal complement spanned by a harmonic cubic f : fthen has two orthogonal isotropic subspaces, U and U0. This establishes a rationalmap from Z to P3 = P(U⊥

0 ), but we must examine the indeterminacy.

Proposition 8.1. If dim(U + U0) < 6 then it is defined by an icosahedral setwith a point in common with [a1], [a2], . . . , [a6].

Proof. Suppose that dim(U + U0) < 6, then dim(U + U0)⊥ ≥ 2 and there

is a pencil of cubics passing through the two icosahedral sets. Suppose that theicosahedral set defined by U is nondegenerate: [b1], [b2], . . . , [b6]. If two axes of anicosahedron coincide then so do all of the axes, so if the icosahedra are distinctthere must be at least 11 points for the pencil to pass through, but by Bezout’stheorem, unless there is a common component in the pencil, the maximum numberof intersections is nine : we deduce that there must be a common component.

If it is a line, then since no three of the points [ai] or [bi] are collinear thereis a maximum of four – two from each set – on the line. But then we have sevenremaining for the pencil of conics. By Bezout again we reach a contradiction. If thecommon component is a conic, then it contains at most five points from each set,and the remaining point must lie in a pencil of lines and so for some i, j, [ai] = [bj ].The equation is therefore of the form

(8.1) (c, x)((a, x)2 − 1

5(a, a)(x, x))

where [a] = [ai] = [bj ] and (c, a) = 0.If f is orthogonal to a degenerate isotropic subspace then the cubic is tangential

to the conic Q at a point [a] and meets the tangent line at a point [b]. The pencilthus consists of the pencil of planes in P3 containing the tangent line to the rationalcurve R ∼= Q at the corresponding point p. Unless the tangent line is containedin the Clebsch cubic S, the generic curve in the pencil is smooth at p, and we canapply the Cayley-Bacharach theorem (in the degenerate case as in [9] p.672) todeduce that any cubic which passes through the six points [b], [a2], [a3], . . . , [a6] and

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10 NIGEL HITCHIN

is also tangent to Q at [a] must pass through [a1]. But take the cubic which is theconic through [a2], [a3], . . . , [a6] together with the tangent at [a]:

(a, x)((a1, x)2 − 1

5(a1, a1)(x, x)) = 0.

If a1 lies on this then (a, a1) = 0. Repeating for the other [ai] we deduce that theyare all collinear, which is a contradiction.

The tangents to R which are contained in S are the lines Ei, and the pencil isthe same as (8.1). �

For each vertex ai of an icosahedron, Proposition 8.1 has highlighted the roleof the icosahedra with this as a common vertex. They form an orbit of the group ofrotations fixing ai – a C

∗ orbit in Z. Its closure is a rational curve with two extrapoints, fixed by the action, which lie in the divisorD (they are two type 1 degenerateicosahedra, determined by the two tangent lines to Q from [b] = [ai]). These sixrational curves Ci all pass through the basepoint U0 in Z. From Proposition 8.1there is a well-defined map from the complement of these six curves to P3:

β(U) = (U + U0)⊥.

We shall extend this to Z by blowing up certain subvarieties.(i) First we need to deal with the basepoint U = U0. Blow up the basepoint in Z,replacing it by the projectivized tangent space. Since the stabilizer of U0 is finite,the tangent space is naturally so(3). For each b ∈ so(3) ∼= C

3 we orthogonallyproject fb onto U⊥

0 to get pb. If this projection is zero then (f, fb) = f(b) = 0 forall f ∈ U0, so that any cubic which vanishes at the ai also vanishes at b. However,these cubics define a projective embedding of the blown-up plane, and so mustseparate points. It follows that the projection is zero if and only if [b] = [ai] forsome 1 ≤ i ≤ 6. We can therefore extend β to P2 \ {[a1], [a2], . . . , [a6]} by usingthe orthogonal projection pb.(ii) The six proper transforms of the curves Ci meet the exceptional divisor in thepoints [ai] and we need to blow up these curves to extend the map. Consider pb(t)for b(t) = ai + tv as t → 0, where (v, ai) = 0. We have

fb(t) = fai+ 3t(v, x)

((ai, x)

2 − 1

5(ai, ai)(x, x)

)+ . . .

Now failies in U0, so

pb(t) = 3t(v, x)

((ai, x)

2 − 1

5(ai, ai)(x, x)

)+ . . .

and the coefficient of t extends β to the blow-up of the projectivized tangent spaceat [ai]. Its image in P3 is the Clebsch cubic. The exceptional curve Ei obtained byblowing up [ai] maps to the pencil in Proposition 8.1.

Denote by Z the blown-up Mukai-Umemura threefold.

Remark 8.2. Blowing up Ci picks out a cubic of the form

(8.2) (c, x)((ai, x)2 − 1

5(ai, ai)(x, x))

with (c, ai) = 0 which extends the map β. The normal bundle of Ci is trivial so theblow-up is Ci ×P1 ∼= P1 ×P1 and β collapses the first factor. This corresponds tothe fact that the cubic (8.2) contains the one-parameter family of icosahedral sets

80


which include [ai]. In general the positive-dimensional fibres of the map β give uscubics containing infinitely many icosahedral sets.

8.2. The trisecant surface. The image β(D) ⊂ Z in P3 consists of thosepolynomials f ∈ U⊥

0 which are also orthogonal to a degenerate isotropic subspace.From Proposition 6.1 this is given by the vanishing of an A5-invariant polynomialof degree 10, the restriction of the SO(3,C)-invariant polynomial Δ.

On the other hand, we know that if f is orthogonal to a degenerate isotropicsubspace, the cubic curve C is defined by a plane section of S which is tangent tothe rational curve R. This means that [f ] ∈ P3 lies on the polar line of a tangent

line to R. From the special properties of the curves R and R observed in Section7 this is a trisecant of R. So the degree 10 polynomial vanishes on the trisecantsurface T of R.

Now a generic line meets d trisecants of R if its polar line meets d tangentsto R. But this surface is the tangent developable surface and we can calculate thedegree easily. If X is a curve of degree n and C2 ⊂ H0(P3,O(1)) is the pencil ofplanes through a line then under the one-jet evaluation map

ev : H0(P3,O(1)) → J1(X,OP3(1))

we have a section of

Λ2(J1(X,OP3(1))) ∼= KX(2)

and the degree of this is 2g − 2 + 2n. It vanishes at the points of intersection of and the surface. In our case R is of degree 6 and rational so the developable hasdegree 0− 2 + 12 = 10.

Thus the trisecant surface, which is irreducible, is of degree 10 and given bythe vanishing of the invariant polynomial Δ of Proposition 6.1. It is the locus ofcubics which pass through a nondegenerate icosahedral set [a1], [a2], . . . , [a6] and adegenerate one.

8.3. The three cases for P3. Consider now the trichotomy in the light of theconstruction of the map β : Z → P3. The trisecant surface T = β(D) consists ofcubics which contain a nondegenerate icosahedral set and a degenerate one. Theycontain therefore two or infinitely many. The image of the six blown-up curves Ci

have infinitely many. This image moreover lies in the Clebsch cubic. There onlyremains the rest of the Clebsch cubic, for which there is either one icosahedral setor infinitely many. The whole surface is therefore the locus of sections of E∗ in 4which have a degenerate zero set – the second and third cases of the trichotomy.

We wish finally to determine which cubic curves contain infinitely many icosa-hedral sets, that is, which of the sections of E∗ on Z vanish on a positive-dimensionalvariety Y ⊂ Z. Since c1(Z) > 0, Y must intersect the anticanonical divisor D non-trivially, and so we are looking for [f ] lying in S and also in the trisecant surfaceT .

Now the special property of R we observed is that the polar of any point on it isa tritangent plane of R. This means that [f ] ∈ R defines a plane section of S ⊂ P3

which meets the curve R tangentially at three points. In other words the associatedplane cubic C passes through three and hence infinitely many icosahedral sets. Atrisecant line of R meets the cubic surface S generically in three points, which arethe three points of intersection with R. Thus for these lines T intersects S only inR.

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12 NIGEL HITCHIN

There are degenerate cases – twelve points (an A5-invariant set) on R wherethe tritangent plane to R meets it in two points with multiplicity three instead ofthree with multiplicity two. The trisecant line to R at these points lies entirely inthe Clebsch cubic – these are the six exceptional divisors E1, E2, . . . , E6.

Set-theoretically, T intersects S in R and the six lines. We have thereforeshown:

Theorem 8.3. A generic point of P3 defines a cubic which contains preciselytwo icosahedral sets. If it contains one or infinitely many it lies on the Clebschcubic surface S and in the latter case it lies on the degree six rational curve R ⊂ Sor on the six lines E1, . . . , E6.

Remark 8.4. In terms of divisor classes let H be the pull-back of the hy-perplane divisor on the original projective plane then a plane cubic C through[a1], . . . , [a6] lifts to a curve whose divisor is 3H − (E1 + · · ·+E6). The embeddingfor a cubic surface in P3 is the anticanonical embedding so

−KS ∼ 3H − (E1 + · · ·+ E6).

The divisor of R, the lift of a conic in P2, is 2H.Now consider the other six exceptional curves coming from the conics passing

through five of the six points. These have classes E1 ∼ 2H − (E2 + · · · + E6) etc.and

−KS ∼ 3H − (E1 + · · ·+ E6) = 3H − (E1 + · · ·+ E6)

It follows that H ∼ 5H − 2(E1+ · · ·E6), where H is the hyperplane divisor for P2.

Then R ∼ 2H ∼ 10H − 4(E1 + · · ·E6), so the divisor class of T on S is

−10KS ∼ 10(3H − (E1 + · · ·+ E6)) = 3(10H − 4(E1 + · · ·E6)) + 2(E1 + · · ·E6).

We may also note here that

R+ R ∼ 2H + 10H − 4(E1 + · · ·E6) = 4(3H − (E1 + · · ·+ E6)) ∼ −4KS

so that there is an invariant quartic surface which vanishes on the pair of curves.This is 9σ2

2 − 20σ4.

9. The trichotomy for P(V )

Since a generic f ∈ V defines a cubic which contains a finite number of icosa-hedral sets, and since the stabilizer of an icosahedron is finite, then P3 ⊂ P(V )sweeps out an open set under the action of SO(3,C). In principle, therefore, thetrichotomy for P(V ) entails looking for a hypersurface which intersects P3 in theClebsch cubic. However, the Hilbert polynomial for the four-dimensional represen-tation space of A5 is

1 + t10

(1− t2)(1− t3)(1− t4)(1− t5)

and for the SO(3,C) invariants in the seven-dimensional representation space

1 + t15

(1− t2)(1− t4)(1− t6)(1− t10)

so there is no degree 3 invariant for SO(3,C). However, as shown in [12], there is aninvariant sextic hypersurface in P(V ) which meets P3 tangentially in the Clebschcubic. The formula given in [12] is adapted to the context of spherical harmonicsand functions on the two-sphere, but here we shall adopt the point of view in [13]

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instead, using the realization of the representation space V as S6 – homogeneoussextic polynomials in (z1, z2).

In inhomogeneous form the sextic is written as

u0x6 + u1x

5 + · · ·+ u6

with roots x1, x2, . . . , x6. Writing (ij) for xi − xj the discriminant is the degree 10invariant

Δ = u100

∏i<j

(ij)2.

As we have seen, the vanishing of this describes the locus of cubics f which areorthogonal to a degenerate isotropic subspace. In [13], Igusa defines similar invari-ants, summing over permutations:

A = u20

∑fifteen

(12)2(34)2(56)2

B = u40

∑ten

(12)2(23)2(31)2(45)2(56)2(64)2

C = u60

∑sixty

(12)2(23)2(31)2(45)2(56)2(64)2(14)2(25)2(36)2

and introduces rational invariants J2, J4, J6, J8, J10 where

J6 =A3

221184+

5AB

13824− C

576

and J10 = 2−12Δ.

Theorem 9.1. A generic point of P(V ) defines a cubic which contains preciselytwo icosahedral sets. It contains one or infinitely many if and only if it lies on thehypersurface J6 = 0.

Proof. The proof consists of using the explicit forms in Section 4 of [13] todetermine a degree six invariant which vanishes on the SO(3,C) transforms of the

rational curves R and E1, . . . , E6. This can be done by evaluating on the normalforms below:(i) A point in R gives a sextic polynomial in z with three double zeros, and eachsuch is equivalent under the action of SO(3,C) to z2(z − 1)2.(ii) The cubic polynomials in Ei are of the form

(c, x)((ai, x)2 − 1

5(ai, ai)(x, x))

with (c, a) = 0. The line (a, x) = 0 intersects Q given by (x, x) = 0 at two pointsin general and c is a third point on this line. Its polar line (c, x) = 0 intersects Qin two more points, so the sextic has two double zeros and two simple ones if [c]does not lie on Q. There is a constraint however – identifying Q with P1 the fourpoints must have cross-ratio −1. A normal form for this is z2(z2 − 1).

We finally need to consider the points of P(V ) whose SO(3,C) orbits do notintersect P3. This means cubics which do not contain any nondegenerate icosahe-dral set. If the cubic contains two degenerate icosahedral sets, then it is tangentialtwice to the null conic, so in the sextic polynomial interpretation we can take it tothe form

z2(az2 + bz + c).

83

14 NIGEL HITCHIN

and evaluating J6 gives

1

1024b2(b2 − 4ac)2.

If J6 = 0, then either b2 − 4ac = 0 which is Case (i) above or b = 0 which is Case(ii). �

10. A special cubic curve

We remarked above that the sextic polynomial z2(z − 1)2 is a normal form foran element in S6 which represents a cubic curve containing an infinite number oficosahedral sets. There is thus a single SO(3,C) orbit for a cubic which is tangentialto the null conic at three distinct points. Perhaps the simplest canonical form forthis is

y2 = x3 + x2 + 4x+ 4

where the null conic has equation

y2 = 4(2x− 3)(x+ 1).

The two meet at just three points: (4, 10), (4,−10), (−1, 0).Note that this cubic C is defined over the integers and has rational points

∞, (4,−10), (0,−2), (−1, 0), (0, 2), (4, 10)

forming a cyclic group of order six.Now C contains a one-parameter family of icosahedral sets, three of which

are degenerate. Moreover this family is rational. To see this, we revert to theisotropic subspace description. From Proposition 8.1 we may assume that there aretwo isotropic subspaces U1, U2 of W for which U1 ∩ U2 = 0. A third subspace U3

intersects each of these trivially and hence is the graph of a linear transformationS : U1 → U2. Thus for w1, w2 ∈ U1 we have

0 = ωx(w1 + Sw1, w2 + Sw2) = ωx(w1, Sw2) + ωx(Sw1, w2)

since ωu(w1, w2) = 0 = ωu(Sw1, Sw2). This means the graph of tS for any t ∈ C isalso isotropic, so we have a family of isotropic subspaces with rational parameter t.

Each icosahedral set consists of six points and so we have a sixfold coveringp : C → P1 with three branch points 0, 1,∞ in P1. At these points the icosahedrondegenerates – one axis remains (the vector b in the type 1 degenerate subspace)and the other five coalesce to be a null vector (the vector a). The inverse image ofeach branch point consist of two points – one, [a], a ramification point of order 5and one simple point [b]. These six points are precisely the six rational points inthe canonical form above.

The projection map p is given in its simplest form by the meromorphic function

p2 =(1 + x)5(y − x− 2)

(y + 3x− 2)5.

An explicit form for p is a little less simple – it involves elliptic functions for twoplanar embeddings (see Remark 5 in Section 12), which differ by the line bundle Lof order 2 , for here we are considering a curve where the vector bundle is L ⊕ Lwith L2 trivial.

84


Remark 10.1. The formulae above have come from [17] in the context ofalgebraic solutions to the hypergeometric equation. The monodromy of the coveringp lies in the icosahedral group and mapping into PSL(2,C) this gives a secondorder differential equation with three singular points. Schwarz’s celebrated paper[16] classifying these stops short of explicit expressions and the elliptic curve aboveappears in an attempt to be more concrete about these.

11. Higher degree curves

The picture for higher degree representations and curves is not clear – in par-ticular there are stable bundles and not just semi-stable as in the case of the ellipticcurve. One might ask for a link between semistability and isotropic subspaces, how-ever. The argument in Section 3 generalizes to show that a d-dimensional isotropicsubspace orthogonal to a harmonic polynomial f of degree d defines a line subbun-dle L with degL = degE/2, though it is not clear if the reverse is true. What istrue is that isotropic subspaces are rare. We can find some, though, as we showbelow.

Let V now be the (2d + 1)-dimensional irreducible representation space ofSO(3,C) and consider a d-dimensional subspace isotropic for all skew forms (x·u, v)which is invariant by SO(2,C) ⊂ SO(3,C). Then it is a sum of d weight spaces Lm

where −d ≤ m ≤ d. Let h,n+,n− be a basis of so(3) where h generates SO(2,C)and [h,n+] = n+, [h,n−] = −n−.

Proposition 11.1. Let U ⊂ V be a d-dimensional SO(2,C)-invariant isotropicsubspace of V . Then U is a sum of weight spaces ±{1, 2, 3, . . . , d} or ±{0, 2, . . . , d}

Proof. Because h acts as the scalar m on Lm, if a weight m > 0 occurs, then−m does not because on Lm ⊕ L−m, (h · u, v) = muv which is non-zero.

Now n+ : Lm �→ Lm+1, so on Lm ⊕L−m−1, (n+ · u, v) = uv which is non-zero.

Hence if m occurs, −m− 1 does not. Similarly with n−, if m occurs −m+ 1 doesnot.

Suppose 0 is not one of the d weights and let P be the positive weights. Thenthe negative weights N must be the negatives of the complement of P in {1, . . . , d}by the first criterion. If m ∈ P then by the second criterion, the adjacent numbersm− 1,m+ 1 must be in P . But this means that P consists of 1, 2, 3, . . . , d. If P isempty, then we get −1,−2,−3, . . . ,−d

If 0 is one of the weights, then by the second criterion ±1 is not a weight. ThenP is a subset of 2, . . . , d. Again the d− 1 non-zero weights must be made up of Pand the negative of its complement. As before adjacent positive numbers must bein P , so P is 2, 3, . . . , d so the weights are 0, 2, . . . , d or their negatives. �

Note that these are precisely the degenerate isotropic subspaces which we haveencountered for d = 3. So if the curve f(x) = 0 has a high order contact with theconic Q, we may deduce that the bundle E is strictly semistable.

12. Further remarks

1. We have shown here that a generic cubic which passes through one icosahedralset passes through another one. According to Melliez [14] (who also introducedthe use of the bundle E∗), this is a result of Reye and can be found in Baker [2],page 145, Exercise 26 or Coble [6] page 236. The reader is invited to make thetranslation.

85

16 NIGEL HITCHIN

2. A discussion of the real case can be found in [12]. There we consider the picturein the sphere double covering RP2. The inverse image of the cubic curve is thenthe so-called nodal set of a spherical harmonic and the issue is whether the nodalset contains the vertices of a regular icosahedron. The rational curves R and R arenot real and so do not appear in the story.

3. The curve R is simply the null conic in P2 but is much more complicated whenviewed in P2. We saw from Remark 8.4 that its divisor class is 10H − 4(E1 +· · ·E6) so it is of degree 10. In fact it is the invariant singular curve defined by:2(x1

10+x310 +x2

10)+ 35(x18x2

2+x12x2

8+x22x3

8+x28x3

2+x38x1

2+x32x1

8)+

25√5(x1

2x28+x2

2x38+x3

2x18−x1

8x22−x2

8x32−x3

8x12)−30(x1

6x24+x1

4x26+

x26x3

4 + x24x3

6 + x36x1

4 + x34x1

6) + 50√5(x1

6x24 + x2

6x34 + x3

6x14 − x1

4x26 −

x24x3

6−x34x1

6)−560x21x

22x

23(x

41+ +x4

2+x43)+1060x2

1x22x

23(x

21x

22+x2

2x23+x2

3x21) = 0.

4. The closure of the curves in the Mukai-Umemura threefold defined by icosahedrawith a fixed axis are rational curves C with c1(Z)[C] = 2, i.e degree 2. These are theminimal rational curves appearing in the classification of Fano varieties. The one-parameter family of icosahedra parametrized by points of R define degree 3 curves,for the plane cubic C is tangential to the null conic at three points. Whereas degree 3rational curves were linked to icosahedral solutions to the hypergeometric equation,degree 4 curves give rise to algebraic solutions of a particular case of Painleve’s sixthequation. This is the setting of [10],[11] where the dihedral and octahedral casesare examined. A classification of all such solutions, akin to Schwarz’s list, has beengiven by Boalch [4].

To find degree 4 rational curves we consider the map β : Z → P3 of Section8.1. Since R parametrizes cubics containing an infinite number of icosahedral sets,the map β collapses a subvariety in Z. If we want to go the other way, from P3

to Z, we first blow up R and then do some blowing up and down. Now take asecant to the curve R ⊂ P3 – a line joining two points. A generic line intersects thetrisecant surface T in 10 points. However, the 3 : 3 correspondence between R andR implies that through a generic point of R there pass three trisecants. Blowingup R means that the proper transform of the secant is a curve which meets theblown-up trisecant surface in 10 − 3 − 3 = 4 points, and this becomes a rationalcurve of degree 4 in the Mukai-Umemura threefold. The corresponding solutionto Painleve VI is known and is due to Dubrovin and Mazzocco [8]. An open setin the Mukai-Umemura threefold can then be identified with the twistor space ofa complex self-dual four-manifold and the secants above, after transforming bySO(3,C), give the four-parameter family of twistor lines.

5. One might ask, without reference to the vector bundle E, how a cubic curvedefined by a harmonic polynomial gives rise to a line bundle L. It arises from thegeometry as follows.

As a varies in the cubic C consider the polynomial fa, invariant under the actionof a ∈ so(3). Now ωa(fa, v) = (a · fa, v) = 0 so that fa generates a line bundle inthe kernel of ω : W → W ∗(1). We identify this bundle as follows. Consider themap γ : P2 → P(V ) defined by a �→ fa. Then fa spans the pull-back γ∗(O(−1))and since fa is homogeneous of degree 3 in a, γ∗(O(−1)) ∼= OP2(−3). Restrict thisto C to get the required line bundle.

We then have an inclusion O(−3) ⊂ E∗(−1) over C. Projecting toE∗(−1)/L(−1) gives a section of L∗(2). This vanishes when fa ∈ U which is

86


at the six points [a1], . . . , [a6] defined by the icosahedral set. Thus on C we havethe relation of divisor classes

L ∼ O(2)−6∑1

[ai].

Now we saw that the cubics through the icosahedral set are given by plane sectionsof the Clebsch cubic surface. This means that the same cubic is embedded in twodifferent planes – a plane in P3 and the original plane with the conic Q. Since theembedding in P3 is

O(3)−6∑1

[ai]

we see that the degree zero line bundle L is the difference of the two hyperplanedivisor classes.

References

[1] M.F.Atiyah, Vector bundles over an elliptic curve, Proc. London Math. Soc. 7 (1957), 414–452.

[2] H.F.Baker, “Principles of Geometry”, Vol 3 Cambridge Univ. Press, Cambridge (1933).[3] A.Beauville, Determinantal hypersurfaces, Michigan Math. J. 48 (2000) 39–64.[4] P.Boalch, The fifty-two icosahedral solutions to Painleve VI, J. Reine Angew. Math. 596

(2006), 183–214.[5] I. Burban and Y. Drozd, Coherent sheaves on rational curves with simple double points and

transversal intersections, Duke Math. J. 121 (2004), 189–229.[6] A. B. Coble, “Algebraic geometry and theta functions,” 10 AMS Coll. Publ., 1929.[7] S.K.Donaldson, Kahler geometry on toric manifolds, and some other manifolds with large

symmetry, arXiv:0803.0985[8] B. Dubrovin and M. Mazzocco, Monodromy of certain Painleve-VI transcendents and reflec-

tion groups, Invent. Math. 141 (2000), 55–147.[9] P.Griffiths and J.Harris, “Principles of algebraic geometry”, John Wiley & Sons New York,

1978.[10] N.J.Hitchin, Poncelet polygons and the Painleve equations, in “Geometry and analysis (Bom-

bay, 1992)”, 151 – 85, Oxford University Press, Bombay, 1996.[11] N.J.Hitchin, A lecture on the octahedron, Bulletin of the London Math Soc., 35 (2003),

577–600.

[12] N.J.Hitchin, Spherical harmonics and the icosahedron, Groups and symmetries, 215–231,CRM Proc. Lecture Notes, 47, Amer. Math. Soc., Providence, RI, 2009.

[13] J-I. Igusa, Arithmetic variety of moduli for genus two, Ann. of Math. 72 (1960), 612–649.[14] F.Melliez, Duality of (1, 5) polarized abelian surfaces, Math. Nachr. 253 (2003), 55–80.[15] S.Mukai, Fano 3-folds, in “Complex projective geometry (Trieste, 1989/Bergen, 1989)” 255–

263, London Math. Soc. Lecture Note Ser. 179 Cambridge Univ. Press, Cambridge (1992).

[16] H.A.Schwarz, Uber diejenigen Falle, in welchen die Gaussische hypergeometrische Reihe einealgebraische Funktion ihres vierten Elements darstellt, J. Reine Angew. Math. 75 (1872).

[17] R.Vidunas, Darboux evaluations of algebraic Gauss hypergeometric functions, arXiv:math0504264

Mathematical Institute, 24-29 St Giles, Oxford OX1 3LB, UK


87


Cohomology of the Toroidal Compactification of A3

Klaus Hulek and Orsola Tommasi

Abstract. We prove that the cohomology groups with rational coefficients ofthe Voronoi compactification AVor

3 of the moduli space of abelian threefoldscoincide with the Chow groups of that space, as determined by Van der Geer.

1. Introduction

The moduli space Ag of principally polarized abelian varieties has several com-pactifications, notably the Satake compactification ASat

g and various toroidal com-pactifications. Among the toroidal compactifications the so called Voronoi com-pactification AVor

g is distinguished by the fact that it represents a geometricallymeaningful functor, as was shown by Alexeev [A] and Olsson [O]. Toroidal com-pactifications are defined by suitable fans in the cone of semi-positive symmetricreal (g× g) matrices and in the case of AVor

g the fan is given by the second Voronoidecomposition. For a definition of the second Voronoi fan we refer the reader to[V] or for a more modern reference to [AN]. A general discussion of toroidalcompactifications of Ag can be found in the survey article [HS].

In genus 3 all known toroidal compactifications of the moduli space A3 ofprincipally polarized abelian varieties coincide with the Voronoi compactificationAVor

3 . We recall the explicit description of the second Voronoi decomposition in thecase g = 3 in Section 5. A detailed description of the geometry of the space AVor

3

can be found in [Ts], and the Chow ring of this space has been computed by Vander Geer [vdG].

In this note we compute the cohomology groups with rational coefficients andprove that they coincide with the Chow groups of this space.

Theorem 1.1. The Betti numbers of AVor3 are b0 = b12 = 1, b2 = b10 = 2,

b4 = b8 = 4 and b6 = 6.

Our approach is similar to that of [vdG], and is based on a study of thestratification of AVor

3 defined by the torus rank, which we introduce in Section 2.1.

2010 Mathematics Subject Classification. Primary 14K10; Secondary 14C15, 14F25, 14D22.Key words and phrases. Abelian varieties, Voronoi compactification, Chow ring, cohomology

ring.Partial support from DFG under grant Hu/6-1 is gratefully acknowledged. We are also

grateful to the referee for careful reading and valuable suggestions concerning the presentation.


1



89

2 KLAUS HULEK AND ORSOLA TOMMASI

We shall give the proof of the main result in Section 2.4 modulo the computationof the cohomology of the various strata, which will be done in the subsequentsections.

As a corollary we obtain

Corollary 1.2. The cycle map defines an isomorphism

CH•(AVor3 )⊗Q ∼= H•(AVor

3 ;Q)

between the Chow ring and the cohomology ring of AVor3 with rational coefficients.

Proof. The Betti numbers coincide with the rank of the Chow groups as de-termined by Van der Geer ([vdG]). Since the intersection pairing is non-degenerate,the cycle map gives an isomorphism. �

Although this result is not particularly surprising, we could not find a referenceto it in the literature, so we decided to fill the gap with this note.

We would like to remark that there are other possible approaches that yieldthe same result. For instance, one can consider the Torelli map M3 → AVor

3 fromthe moduli space of Deligne–Mumford stable curves of genus 3 to the toroidalcompactification of A3. The moduli space M3 has a stratification by topologicaltype. Since the Torelli map for genus 3 is surjective, we can stratify AVor

3 by takingthe images of the strata of M3. It is easy to show that all strata of AVor

3 obtainedin this way are isomorphic to finite quotients of products of moduli spaces Mg,n

with g ≤ 3 and 0 ≤ n ≤ 2(g − 3). Then one can use the known results about thecohomology of these spaces Mg,n to calculate the cohomology of AVor

3 .In this note, we will work with the stack AVor

3 rather than the associated coarsemoduli space. We recall that AVor

3 is a smooth Deligne–Mumford stack. Hence therational cohomology of the stack and the associated coarse moduli space coincide.

Finally we remark that the same techniques also apply to the (easier) case ofgenus 2.

Remark 1.3. There is an isomorphism

CH•(AVor2 )⊗Q ∼= H•(AVor

2 ;Q).

Notation.

Ag moduli stack of principally polarized abelian varieties of genus g

Xg universal family over Ag

ASatg Satake compactification of Ag

AVorg Voronoi compactification of Ag

XVorg universal family over AVor

g

Mg,n moduli stack of non-singular curves of genus g with n markedpoints

Mg := Mg,0

Sd symmetric group in d letters

For every g, we denote by ϕg : AVorg → ASat

g the natural map from the Voronoi

to the Satake compactification. Let πg : XVorg → AVor

g be the universal family,

90

COHOMOLOGY OF THE TOROIDAL COMPACTIFICATION OF A3 3

qg : XVorg → XVor

g /± 1 the quotient map from the universal family to the universal

Kummer family and kg : XVorg /± 1 → AVor

g the universal Kummer morphism.Throughout the paper, we work over the field C of complex numbers.

2. Stratification and outline of the proof

2.1. A stratification. The object of this note is the rational cohomologyof the toroidal compactification AVor

3 of the moduli space of abelian varieties ofdimension 3. We shall make use of a natural stratification of AVor

3 which was alsoused by Van der Geer [vdG], whose notation we adopt.

Recall that there is a natural map ϕ3 : AVor3 → ASat

3 to the Satake compactifi-cation. The moduli space ASat

3 admits a stratification ASat3 = A3 � A2 � A1 � A0.

This defines a filtration {βt}0≤t≤3 on AVor3 , by setting

βt := ϕ−13

⎛⎝ ⊔

0≤j≤g−t

Aj

⎞⎠ .

In other words, βt ⊂ AVor3 is the locus of semi-abelian varieties with torus rank at

least t.

2.2. Cohomology of the strata. We shall now state the results about thecohomology with compact support of the various strata. Proofs will be given in thesubsequent Sections 3 – 5.

The stratum β0 \ β1 of the filtration {βt} is A3. Its cohomology was computedby Hain in [H].

Theorem 2.1. The rational cohomology groups with compact support of A3 aregiven by

Hkc (A3;Q) =

⎧⎪⎪⎪⎪⎨⎪⎪⎪⎪⎩

Q(−6) k = 12,Q(−5) k = 10,Q(−4) k = 8,F k = 6,0 otherwise,

where F is a two-dimensional mixed Hodge structure which is an extension

0 → Q → F → Q(−3) → 0.

Proof. This is a rephrasing of [H, Thm 1], by using the isomorphism

Hkc (A3;Q) ∼= H12−k(A3;Q)∗ ⊗Q(−6)

given by Poincare duality on the 6-dimensional space A3. �

The cohomology with compact support of the other strata is as follows.

Proposition 2.2. The cohomology with compact support of β1 \ β2 is given by

H10c (β1 \ β2;Q) = Q(−5)

H8c (β1 \ β2;Q) = Q(−4)2

H6c (β1 \ β2;Q) = Q(−3)2

H5c (β1 \ β2;Q) = Q

H4c (β1 \ β2;Q) = Q(−2)

Hkc (β1 \ β2;Q) = 0 for k /∈ {4, 5, 6, 8, 10}.

For torus rank 2 we obtain

91


Proposition 2.3. The cohomology with compact support of the stratum β2 \β3

is given by

H8c (β2 \ β3;Q) = Q(−4)

H6c (β2 \ β3;Q) = Q(−3)2

H4c (β2 \ β3;Q) = Q(−2)

H2c (β2 \ β3;Q) = Q(−1)

Hkc (β2 \ β3;Q) = 0 for k /∈ {2, 4, 6, 8}.

In the proof of the two propositions above, we make use of the following fact(see [Ts]): the natural map β1 → ASat

2 factors through AVor2 , giving rise to the

commutative diagram

β1k2 ��

π2

��

��

� AVor2

ASat2

��ϕ2

where k2 : β1∼= (XVor

2 /± 1) → AVor2 is the universal Kummer variety over AVor

2 .Finally, we use the toroidal description of AVor

3 to compute the cohomology ofthe stratum with torus rank 3. The corresponding result is

Proposition 2.4. The cohomology groups Hkc (β3;Q) are trivial for degree k /∈

{0, 2, 4, 6}, and are given in the other cases by

H6c (β3;Q) = Q(−3) H2

c (β3;Q) = Q(−1)H4

c (β3;Q) = Q(−2)2 H0c (β3;Q) = Q.

Moreover, the generators of these cohomology groups with compact support can beidentified with the fundamental classes of the strata of AVor

3 corresponding to the

cones σ(3)local, σ

(4)I , σ

(4)II , σ

(5) and σ(6) (to be defined in Section 5).

2.3. Spectral sequences in cohomology. Our proofs of results on the coho-mology of AVor

3 and its strata are based on an intensive use of long exact sequencesand spectral sequences in cohomology with compact support. We shall recall thedefinition of the sequences we use most often in the proofs.

Since the cohomology with rational coefficients of a Deligne–Mumford stackcoincides with that of its coarse moduli space, in this section we will work withquasi-projective varieties. A more stack-theoretical approach can be obtained byrecalling that AVor

g is the finite quotient of the fine moduli scheme AVorg (n) of

abelian varieties with level-n structure for n ≥ 3. Then the same constructions canbe obtained by working on AVor

g (n) equivariantly.Recall that if X is a quasi-projective variety and Y a closed subvariety of X,

then the inclusion Y ↪→ X induces a Gysin long exact sequence in cohomology withcompact support:

· · · → Hk−1c (Y ;Q) → Hk

c (X \ Y ;Q) → Hkc (X;Q) → Hk

c (Y ;Q) → · · ·

By functoriality of mixed Hodge structures ([PS, Prop. 5.54], this exact se-quence respects mixed Hodge structures.

Next, assume we have a filtration ∅ = Y0 ⊂ Y1 ⊂ Y2 ⊂ · · · ⊂ YN = X by closedsubvarieties of X. In this case, there is a spectral sequence Ep,q

r ⇒ Hp+qc (X;Q) as-

sociated to the filtration {Yi}. The E1 term is given by Ep,q1 = Hp+q

c (Yp \Yp−1;Q).

This spectral sequence can be constructed by taking a compactification X of X

92


with border S := X \ X. Let us denote by Yi the closure of Yi in X, and con-sider the filtration {Y ′

i := Yi ∪ S}0≤i≤N of the pair (X,S). In particular, one hasH•(Y ′

j , Y′j−1;Q) = H•

c (Yj \ Yj−1;Q) for all j ≥ 1. One can describe the spectralsequence associated to {Yi} as the spectral sequence associated to the bigraded ex-act couple (D,E) with Dα,β = Hα+β(X,Y ′

α−1;Q) and Eα,β = Hα+β(Y ′α, Y

′α−1;Q),

which converges to Hα+βc (X,S;Q) = Hα+β(X;Q). Arguing as in [Ar, Lemma 3.8],

this ensures the compatibility with mixed Hodge structures by functoriality. Forthe definition of exact couples, see [PS, §A.3.2].

Note that the d1 differentials of the spectral sequence in cohomology with com-pact support associated to {Yi} coincide with the differentials of the Gysin longexact sequences associated to the closed inclusions Yi \ Yi−1 ↪→ Yi+1 \ Yi−1.

Leray spectral sequences play an intensive role in our computation of the coho-mology of the strata β1 and β2 \β1. Typically, we will be in the following situation:let X and Y be quasi-projective varieties, and f : X → Y a fibration with fibreswhich are homotopy equivalent under proper maps to a fixed quasi-projective vari-ety B. Let us denote by H(p) the local system on X induced by the pth cohomologygroup with compact support of the fibre of f .

In this situation, one can consider the Leray spectral sequence of cohomol-ogy with compact support associated to f . This is the spectral sequence Ep,q

r ⇒Hp+q

c (X;Q) with Ep,q2

∼= Hp(Y ;H(q)). Note that the Leray spectral sequence as-sociated to f respects Hodge mixed structures (e.g. see [PS, Cor. 6.7]).

2.4. Proof of the main theorem. The results on the cohomology with com-pact support stated in Section 2.2 enable us to compute the cohomology of AVor

3

using the spectral sequence Ep,q• ⇒ Hp+q

c (AVor3 ;Q), Ep,q

1 = Hp+qc (β3−p \ β4−p;Q)

associated to the filtration β3 ⊂ β2 ⊂ β1 ⊂ β0 = AVor3 .

Lemma 2.5. The E1 term of the spectral sequence in cohomology with compactsupport associated to the filtration β3 ⊂ β2 ⊂ β1 ⊂ β0 = AVor

3 is as given in Table 2.

The only non-trivial differential of this spectral sequence is d2,31 : E2,31 → E3,3

1 , whichis injective. In particular, the spectral sequence degenerates at E2.

Proof. The description of the E2 term of the spectral sequence follows fromthe description of the compactly supported cohomology of the strata given in Sec-tion 2.2 and from the definition of the spectral sequence in Section 2.3.

An inspection of the spectral sequence in Table 2 yields that Ep,q1 (and hence

Ep,qr ) is always trivial if p+q is odd, with the exception of E2,3

1 (hence possibly alsoE2,3

r for r ≥ 2). Therefore, all differentials not involving E2,3r terms are necessarily

trivial, since they are maps either from or to 0.This leaves us with only three possibly non-trivial differentials to investigate.

The first two are the differentials d2−r,2+rr : E2−r,2+r

r → E2,3r for r = 1, 2. Note

that in both cases, the Hodge structure on E2,3r is pure of weight 0, whereas the

Hodge structure on E2−r,2+rr is pure of weight 4. Since the weights are different,

the differential d2−r,2+rr can only be the 0 morphism.

Next, we investigate the differential d2,31 : E2,31 → E3,3

1 , which can have rank

either 0 or 1. Assume for the moment that d2,31 is the 0 morphism. Then the spectral

sequence degenerates at E1, so that H5(AVor3 ;Q) ∼= H5

c (AVor3 ;Q) = E2,3

1 = Q holds.This means that the cohomology of AVor

3 in degree 5 is pure of Hodge weight 0.But AVor

3 is a smooth proper stack, being the quotient by a finite group of the stack

93


Table 2. E1 term of the spectral sequence converging toH•

c (AVor3 ;Q) = H•(AVor

3 ;Q)

q

9 0 0 0 Q(−6)8 0 0 Q(−5) 07 0 Q(−4) 0 Q(−5)6 Q(−3) 0 Q(−4)2 05 0 Q(−3)2 0 Q(−4)4 Q(−2)2 0 Q(−3)2 03 0 Q(−2) Q F2 Q(−1) 0 Q(−2) 01 0 Q(−1) 0 00 Q 0 0 0

0 1 2 3 p( torusrank ) (3) (2) (1) (0)

AVor3 (n) of principally polarized abelian varieties with a level-n structure, which is

represented by a smooth projective scheme for n ≥ 3. In particular, the Hodgestructure on Hk(AVor

3 ;Q) is pure of weight k. Hence, the rank of d2,31 must be 1.Therefore, this differential is injective with cokernel isomorphic to Q(−3). This

ensures E2,32 = 0 and E3,3

2 = Q(−3). �Note that Lemma 2.5 directly implies that the cohomology of AVor

3 is all alge-braic, with Betti numbers as stated in Theorem 1.1.

In the remainder of this paper we will discuss the various strata defined by thetorus rank and compute their cohomology.

3. Torus rank 1

To compute the cohomology with compact support of β1\β2 we will use the mapk2 : β1 \ β2 → A2 realizing β1 \ β2 as the universal Kummer variety over A2. Thefibre of β1 \β2 over a point parametrizing an abelian surface S is K := S/±1. Thecohomology of K is one-dimensional in degree 0 and 4. The only other non-trivialcohomology group is H2(K;Q) ∼=

∧2 H1(S;Q).To compute H•

c (β1 \β2;Q), we consider the Leray spectral sequence associatedto k2. Note that the 0th and the fourth cohomology group of the fibre inducetrivial local systems on A2. Moreover, the second cohomology group of the fibreinduces the rank 6 local system V(1,1) ⊕ Q(−1) on A2. Here we denote by V(1,1)

the symplectic local system on A2 determined by the irreducible representation ofSp(4,Q) associated to the partition (1, 1).

We start by determining the cohomology with compact support of A2 withvalues in the local system V(1,1).

Lemma 3.1. The rational cohomology groups with compact support of the modulispaces M2 and A2 with coefficients in V(1,1) vanish in degree k �= 3. In degree 3,one has

H3c (A2;V(1,1)) = H3

c (M2;V(1,1)) = Q.

94


Proof. We prove the claim about the cohomology of M2 first.Following the approach of [G2], we use the forgetful map p2 : M2,2 → M2

to obtain information. Note that the fibre of p2 is the configuration space of 2distinct points on a genus 2 curve. The cohomology of M2,2, with the action ofthe symmetric group, was computed in [T, Cor. III.2.2]. This result allows us toconclude H3

c (M2;V(1,1)) = Q, Hkc (M2;V(1,1)) = 0 for k �= 3. (Note that this is

in agreement with the Hodge Euler characteristic of M2 in the local system V(1,1)

computed in [G2, §8.2].)Next, we determine H•

c (A2;V(1,1)). To this end, we write A2 as the disjointunion of the locus A1,1 of decomposable abelian surfaces, and the image of theTorelli map t : M2 → A2.

Since the Torelli map is injective on the associated coarse moduli spaces, itinduces an isomorphism between the cohomology of M2 and that of t(M2) in everysystem of coefficients that is locally isomorphic to a Q-vector bundle. Therefore, theGysin long exact sequence with V(1,1)-coefficients associated to A1,1 ↪→ A2 yields

Hk−1c (A1,1;V(1,1)) → Hk

c (M2;V(1,1)) → Hkc (A2;V(1,1)) → Hk

c (A1,1;V(1,1)).

In Lemma 3.3 below, we will show that H•c (A1,1;V(1,1)) is trivial. In view of

the Gysin exact sequence above, this implies that Hkc (A2;V(1,1)) is isomorphic to

Hkc (M2;V(1,1)). This implies the claim. �Remark 3.2. Getzler’s result would have been sufficient for the purposes of

this note. This follows again from the fact that AVor3 is a finite quotient of AVor

3 (n),so in particular its Hodge Euler characteristic determines the cohomology of thespace as graded vector space with Q-Hodge structures.

Lemma 3.3. The cohomology with compact support of A1,1 in the local systemof coefficients given by the restriction of V(1,1) is trivial.

Proof. We consider the restriction of k2 to A1,1. Let S = E1 × E2 be an

element of A1,1, and let K := k−12 (S). Recall that V(1,1) ⊕ Q(−1) is the local

system H(2) on A1,1 induced by H2(K;Q). Therefore, the cohomology of A1,1

with values in V(1,1) ⊕Q(−1) coincides with the cohomology of A1,1 with values in

the local system induced by the part of∧2 H1(S;Q) which is invariant under the

symmetries of E1 ×E2 and under the interchange of the two factors E1, E2 (whichcan be done topologically albeit not algebraically). Using the Kunneth formula onesees that the latter local system is one-dimensional and induces the local systemQ(−1). From this one obtains H•

c (A1,1;V(1,1)) = 0. �This allows us to show the following result, which directly implies that the

cohomology with compact support of β1 \ β2 is as stated in Proposition 2.2.

Proof of Proposition 2.2. We compute the cohomology with compact sup-port of β1 \ β2 by using the Leray spectral sequence associated to the Kummerfibration k2 : β1 \ β2 → A2.

By the description of the fibre of k2 given at the beginning of this section, thelocal systems H(0) and H(4) are the constant one, whereas H(2) is the direct sumof the constant local system Q and V(1,1).

The cohomology with compact support of A2 is well known: it is one-dimen-sional in degree 4 and 6, and trivial elsewhere. This can be easily deduced from theresults in [M2] on the Chow ring of M2. The cohomology of A2 in the local system

95


Table 3. E2 term of the Leray spectral sequence converging tothe cohomology with compact support of β1 \ β2

q

4 0 Q(−4) 0 Q(−5)3 0 0 0 02 Q Q(−3) 0 Q(−4)1 0 0 0 00 0 Q(−2) 0 Q(−3)

3 4 5 6 p

V(1,1) was computed in Lemma 3.1. From this, one obtains that the E2 term of theLeray spectral sequence in cohomology with compact support associated to k2 is asin Table 3.

From an inspection of the spectral sequence, one finds that all Ep,q2 have pure

Hodge structures, which have the same Hodge weight if and only if the sums p +q coincide. Therefore, all differentials dr (r ≥ 2) of the spectral sequence aremorphisms between Hodge structures of different weight. Hence all differentials aretrivial for this reason. This means that the spectral sequence degenerates at E2,thus implying Proposition 2.2. �

4. Torus rank 2

Recall that k2 : β1 → AVor2 is the universal family of Kummer varieties over

AVor2 . Under this map, the elements of AVor

3 with torus rank 2 are mapped toelements of AVor

2 of torus rank 1. If we denote by β′t the stratum of AVor

2 of semi-abelian varieties of torus rank ≥ t, we get a commutative diagram

AVor3��

AVor2��

AVor1��

β2 \ β3k2 �� β′

1 \ β′2

k1 �� A1

π−12 (β′

1 \ β′2)

q2

��

π2

��X1

q1

��

π1

��

The map π2 is the restriction of the universal family over AVor2 . In particular,

the fibres of π2 over points of β′1 \ β′

2 are rank 1 degenerations of abelian surfaces,i.e. compactified C

∗-bundles over elliptic curves. A geometric description of theseC∗-bundles is given in [M1].

We want to describe this situation in more detail. For this consider the universalPoincare bundle P → X1 ×A1

X1 and let U = P(P ⊕ OX1×A1X1

) be the associated

P1-bundle. Using the principal polarization we can naturally identify X1 and X1,which we will do from now on. We denote by Δ the union of the 0-section and the∞-section of this bundle. Set U = U \Δ, which is simply the C∗-bundle given bythe universal Poincare bundle P with the 0-section removed and denote the bundlemap by f : U → X1 ×A1

X1. Then there is a map ρ : U → β2 \ β3 with finite fibres.Note that the two components of Δ are identified under the map ρ. The restriction

96


of ρ to both U and to Δ is given by a finite group action, although the group is notthe same in the two cases (see the discussion below).

We now consider the situation over a fixed point [E] ∈ A1. For a fixed degree 0line bundle L0 on E the preimage f−1(E×{L0}) is a semi-abelian surface, namely

the C∗-bundle given by the extension corresponding to L0 ∈ E. This semi-abelian

surface admits a Kummer involution ι which acts as x �→ −x on the base E and byt �→ 1/t on the fibre over the origin. The Kummer involution ι is defined universallyon U .

Consider the two involutions i1, i2 on X1 ×A1X1 defined by i1(E, p, q) =

(E,−p,−q) and i2(E, p, q) = (E, q, p) for every elliptic curve E and every p, q ∈ E.These two involutions lift to involutions j1 and j2 on U that act trivially on thefibre of f : U → X1 ×A1

X1 over the origin.

Lemma 4.1. The diagram

(4.1) U ��

ρ|U��

X1 ×A1X1

ρ′

��(β2 \ β3) \ ρ(Δ) �� Sym2

A1(X1/± 1),

where ρ′ : X1 ×A1X1 → Sym2

A1(X1/± 1) is the natural map, is commutative.

Moreover ρ|U : U → ρ(U) ⊂ β2 \ β3 is the quotient of U by the subgroup of theautomorphism group of U generated by ι, j1 and j2.

Proof. Since the map ρ′ in the diagram (4.1) has degree 8 and ι, j1, j2 generatea subgroup of order 8 of the automorphism group of U , it suffices to show that themap ρ|U factors through each of the involutions ι and j1, j2.

Recall that the elements of β2\β3 correspond to rank 2 degenerations of abelianthreefolds. More precisely, every point of ρ(U) corresponds to a degenerate abelianthreefold X whose normalization is a P1 × P1-bundle, namely the compactificationof a product of two C

∗-bundles on the elliptic curve E given by k1 ◦ k2([X]). Thedegenerate abelian threefold itself is given by identifying the 0-sections and the∞-sections of the P1 × P1-bundle. This identification is determined by a complexparameter, namely the point on a fibre of U → X1 ×A1

X1.Since a degree 0 line bundle L0 and its inverse define isomorphic semi-abelian

surfaces and since the role of the two line bundles is symmetric, the map ρ|U factorsthrough ι andj2. Since j1 is the commutator of ι and j2 the map ρ|U also factorsthrough j1. �

A consequence of the lemma above is that the cohomology with compactsupport of ρ(U) can be computed by taking the invariant part of the cohomol-ogy of the total space of the C∗-bundle f : U → X1 ×A1

X1. Hence, the in-variant part of the Leray spectral sequence associated to f gives a Leray spec-tral sequence converging to H•

c (ρ(U);Q). Thus, we have to consider the part ofEp,q

2 (f) = Hqc (C

∗;Q)⊗Hpc (X1 ×A1

X1;Q) that is invariant under the action of ι, j1and j2.

Since j1 and j2 both fix the fibre of f over the origin, they act trivially onthe cohomology of C∗. Instead, the Kummer involution ι acts as the identity onH2

c (C∗;Q) and as the alternating representation on H1

c (C∗;Q).

The action of ι, j1 and j2 can be determined by considering the induced actionson X1×A1

X1. Here one uses that all three involutions respect the map X1×A1X1 →

97


Table 4. E2 term of the spectral sequence converging to thecohomology with compact support of ρ(U)

q

2 Q(−2) 0 Q(−3) 0 Q(−4)1 0 0 Q(−2) 0 0

2 3 4 5 6 p

A1, whose fibre over [E] ∈ A1 is isomorphic to E × E. Note in particular that theinvolution (E, p, q) ↔ (E,−p, q) induced by ι acts as the alternating representation

on the linear subspace∧2

H1c (E;Q) ⊂ H2

c (E × E;Q), on which i1 and i2 both acttrivially.

This discussion yields that the invariant part of the spectral sequence E2 termis as shown in Table 4.

Lemma 4.2. The cohomology groups with compact support of ρ(U) are 1-dimen-sional in degree 6 and 8 and trivial otherwise.

Proof. It suffices to show that the differential d2,22 : E2,22 → E4,1

2 in Table 4is an isomorphism.

To describe the differential d2,22 geometrically, it is useful to consider the re-striction of the Torelli map t : M3 → AVor

3 to the preimage of ρ(U). Moreover, onecan use the stratification of M3 by topological type to describe β2 and ρ(U). Inparticular, this allows one to find a geometric generator for H4

c (ρ(U);Q).Consider stable curves C1 ∪ C2 ∪ C3, where the component C1 is smooth of

genus 1, the component C2 is a smooth rational curve and the component C3 is arational curve with exactly one node, satisfying #(C1 ∩ C2) = 1, #(C1 ∩ C3) = 0and #(C2 ∩ C3) = 2.

Denote by G the closure in t−1(ρ(U)) of the locus of such curves, and denoteby t∗[G] the push-forward to ρ(U) of the cycle class of G. Then the fundamentalclass of t∗[G] generates H4

c (ρ(U);Q).Recall that the locus in M3 of irreducible curves with two nodes maps sur-

jectively to β2 under the Torelli map. Moreover, all curves in M3 that have twonodes and map to β2 can be constructed by taking a stable curve of genus 1 with 4marked points and identifying the marked points pairwise. There is a well knownrelation between cycle classes of dimension 2 in M1,4, called Getzler’s relation (see[G1]). This relation is S4-invariant and it induces a relation between dimension 2cycles in t−1(β2), which if pushed forward under t induces a relation in H4

c (β2;Q).The latter relation involves non-trivially the push-forward of the fundamental classof G ⊂ t−1(β2). In particular, restricting to ρ(U) ⊂ β2 yields that t∗[G] vanishes

in H4c (ρ(U);Q). Hence, the differential d2,22 must be an isomorphism. �

Remark 4.3. There is also another way to see that the differential d2,22 : E2,22 →

E4,12 in Table 4 is an isomorphism. Namely, one can compactify the C

∗-bundle Uto the P

1-bundle U = P(P ⊕ OX1×A1X1

) and compute the invariant part of the

exact sequence in rational cohomology of the pair (U,Δ). This then shows thatthe invariant part of H4

c (U ;Q) vanishes as claimed. We decided to include the

98


above proof involving Getzler’s relation since the relation to M3 is of independentinterest.

Proof of Proposition 2.3. We compute the cohomology with compact sup-port of β2\β3 by exploiting the Gysin long exact sequence associated to the inclusionρ(Δ) ↪→ (β2 \ β3):(4.2)

· · · → Hk−1c (ρ(Δ);Q) → Hk

c (ρ(U);Q) → Hkc (β2 \ β3;Q) → Hk

c (ρ(Δ);Q) → · · ·The map ρ identifies the two components of Δ, each of which is isomorphic

to X1 ×A1X1. Moreover, it factors through the finite group G generated by the

following three involutions: the involution which interchanges the two factors ofX1×A1

X1, the involution which acts by (x, y) �→ (−x,−y) on each fibre E×E andfinally the involution which acts by (x, y) �→ (x+ y,−y). This can be read off fromthe construction of the toroidal compactification (see [HS, Section I] for an outlineof this construction. Also note that the stratum Δ corresponds to the stratum inthe partial compactification in the direction of the 1-dimensional cusp associatedto a maximal-dimensional cone in the second Voronoi decomposition for g = 2. Adetailed description can be found in [HKW, Part I, Chapter 3]).

Hence

H•c (ρ(Δ);Q) ∼= H•

c (E × E/G;Q)⊗H•c (A

1C;Q).

A straightforward calculation shows that the G-invariant cohomology of E × Ehas rank 1 in even dimension and vanishes otherwise. In particular this quotientbehaves cohomologically like P

2.Since Hk

c (ρ(U);Q) and Hkc (ρ(Δ);Q) both vanish if k is odd, the exact sequence

(4.2) splits into short exact sequences

0 → Hkc (ρ(U);Q) → Hk

c (β2 \ β3;Q) → Hkc (ρ(Δ);Q) → 0.

This implies the claim. �

Remark 4.4. We would like to take this opportunity to correct a slight errorin [vdG, 3.8] where it was claimed that the map ρ factors through Sym2

A1(X1/±1)

rather than through the quotient by G. This, however, does not effect the resultsof [vdG].

5. Torus rank 3

The stratum β3 ⊂ AVor3 lying over A0 ⊂ ASat

3 is entirely determined by the fanof the toroidal compactification. For this we first have to describe the Voronoi fanΣ in genus 3.

Consider the free abelian group L3∼= Z3 with generators x1, x2, x3 and let

M3 = Sym2(L3). Then M3 is isomorphic to the space of 3 × 3 integer symmetricmatrices with respect to the basis xi via the map which assigns to a matrix Athe quadratic form txAx. We shall use the basis of M3 given by the forms U∗

i,j ,1 ≤ i ≤ j ≤ 3 given by

U∗i,j = 2δi,jxixj .

Let Sym≥02 (L3⊗R) be the cone of positive semidefinite forms in M3⊗Q. The group

GL(3,Z) acts on Sym≥02 (L3 ⊗ R) by

GL(3,Z) � g : M �−→ tg−1Mg−1.

99


Letσ(6) := R≥0α1 + R≥0α2 + R≥0α3 + R≥0γ1 + R≥0γ2 + R≥0γ3,

where αi = x2i for all i = 1, 2, 3 and γi = (xj−xk)

2 for {i, j, k} = {1, 2, 3}. Since theforms αj , γi form a basis of M3, this is a basic 6-dimensional cone in Sym≥0

2 (L3⊗R).

The Voronoi fan in genus 3 is the fan Σ in Sym≥02 (L3 ⊗ R) given by σ(6) and

all its faces, together with their GL(3,Z)-translates. We use the notation

σ(6) = α1 ∗ α2 ∗ α3 ∗ γ1 ∗ γ2 ∗ γ3,and similarly for the faces of σ(6).

To describe AVor3 , we have to know all possible GL(3,Z)-orbits of σ(6) and its

faces. An i-dimensional cone corresponds to a (6− i)-dimensional stratum in AVor3 .

Since strata of dimension at least 4 necessarily lie over Al with l ≥ 1, we only needto know the orbits of cones of dimension ≤ 3.

The following lemma can be proved using the methods of [Ts] (see [E, Chap-ter 3]).

Lemma 5.1. There are two GL(3,Z)-orbits of 3-dimensional cones, representedby the cones

σ(3)local = α1 ∗ α2 ∗ α3, σ

(3)global = α1 ∗ α2 ∗ γ3.

The stratum associated to σ(3)local lies over A0, that associated to σ

(3)global lies over A1.

There are two GL(3,Z)-orbits of 4-dimensional cones, given by

σ(4)I = α1 ∗ α2 ∗ α3 ∗ γ1, σ

(4)II = α1 ∗ α2 ∗ γ1 ∗ γ2.

In dimension 5 and 6 there is only one GL(3,Z)-orbit. The strata of all cones ofdimension at least 4 lie over A0.

LetH3 = {τ = (τi,j)1≤i,j≤3 : τ = tτ, Im τ > 0}

be the Siegel upper half plane of genus 3. We consider the rank 6 torus T = T 6

with coordinatesti,j = e2π

√−1τi,j (1 ≤ i, j ≤ 3).

These coordinates correspond to the dual basis of the basis U∗i,j . If σ(l) is an

l-dimensional cone in Σ then, since the fan Σ is basic, it follows that the associatedaffine variety Tσ(l)

∼= Cl × (C∗)6−l. The corresponding stratum in AVor

3 is then aquotient of {(0, 0, 0)}× (C∗)6−l by a finite group. We consider the torus embeddingT ↪→ Tσ(6)

∼= C6, where the latter isomorphism holds since σ(6) is a basic cone of

dimension 6. Let T1, . . . , T6 be the coordinates of C6 corresponding to the basisα1, . . . , γ3. If one computes the dual basis of α1, . . . , γ3 in terms of the dual basisof U∗

i,j , one obtains that the torus embedding T ↪→ C6 is given by

T1 = t1,1t1,3t1,2, T2 = t2,2t2,3t1,2, T3 = t3,3t1,3t2,3,T4 = t−1

2,3, T5 = t−11,3, T6 = t−1

1,2.

Let us start by considering the stratum associated to

σ(3)local = α1 ∗ α2 ∗ α3.

Let S1, S2 and S3 be coordinates corresponding to α1, α2 and α3, and lett2,3, t1,3, t1,2 be as above. Then

Tσ(3)local

∼= C3 × (C∗)3 ⊂ C

6 = Tσ(6)

100


with coordinates S1, S2, S3, t−12,3, t

−11,3, t

−11,2, where the inclusion is defined by consid-

ering σ(3)local as a face of σ(6).

The stratum which we add is {(0, 0, 0)} × (C∗)3 modulo a finite group G =

Gσ(3)local

, namely the stabilizer of the cone σ(3)local in GL(3,Z). In order to understand

the action of the group G explicitly, we recall that it is naturally a subgroup of theparabolic subgroup which belongs to the standard 0-dimensional cusp

P =

⎧⎨⎩⎛⎝ g 0

0 tg−1

⎞⎠ : g ∈ GL(3,Z)

⎫⎬⎭ ∼= GL(3,Z) ⊂ Sp(6,Z).

Lemma 5.2. The stratum associated to σ(3)local is an affine variety Y

(3)local =

(C∗)3/G whose only non-trivial cohomology with compact support is in degree 6.

Proof. Since the stratum associated to σ(3)local is the quotient of the smooth

variety (C∗)3 by a finite group, its cohomology and cohomology with compactsupport are related by Poincare duality. Hence, it suffices to show that the rationalcohomology of the stratum is concentrated in degree 0.

Denote by T 3 the rank 3 torus with coordinates (v1, v2, v3) = (t−12,3, t

−11,3, t

−11,2).

The stratum which we add for σ(3)local is then isomorphic to T 3/G. Since σ

(3)local =

α1 ∗ α2 ∗ α3 with αi = x2i , we see that the group G is the group generated by the

permutations of the xi and the involutions (x1, x2, x3) �→ (ε1x1, ε2x2, ε3x3) withεi = ±1. Note that the element −id acts trivially both on H3 and on M3. Hencethe group G is an extension

1 → (Z/2Z)2 → G → S3 → 1,

where S3 denotes the symmetric group in 3 letters. Next, we have to analyzehow this group acts on H3 and on the torus T 3. The permutation of xi and xj

interchanges τi,k and τj,k but fixes τi,j . HenceS3 also acts as group of permutationson the coordinates of T 3. The action of the involutions generating (Z/2Z)2 can beseen for example from⎛

⎝−1 0 00 1 00 0 1

⎞⎠

⎛⎝τ1,1 τ1,2 τ1,3τ1,2 τ2,2 τ2,3τ1,3 τ2,3 τ3,3

⎞⎠

⎛⎝−1 0 0

0 1 00 0 1

⎞⎠ =

⎛⎝ τ1,1 −τ1,2 −τ1,3−τ1,2 τ2,2 τ2,3−τ1,3 τ2,3 τ3,3

⎞⎠ .

Hence, the involution (x1, x2, x3) ↔ (x1, x2,−x3) induces the involution givenby (v1, v2, v3) ↔ (v−1

1 , v−12 , v3) and similarly for the other involutions. This allows

us to describe the quotient T 3/G explicitly, as given by the image of the map

T 3 ∼= (C∗)3 −→ C4

(v1, v2, v3) �−→ (u1 + u2 + u3, u1u2 + u1u3 + u2u3, u1u2u3, u4) = (s1, s2, s3, t),

where

u1 = v1+1

v1, u2 = v2+

1

v2, u3 = v3+

1

v3, u4 =

(v1 −

1

v1

)(v2 −

1

v2

)(v3 −

1

v3

).

Then the image is the hypersurface W ⊂ C4 given by

t2

4− (

s32

+ 2s1)2 + (s2 + 4)2 = 0.

Note that W is a cone with vertex the line t = s32 +2s1 = s2+4 = 0 in C4 over

a plane projective conic. Then the claim follows from the contractibility of W .

101


Alternatively, one can also show that the cohomology H•(T 3/G;Q) is concen-trated in degree 0, by proving that the only cohomology in H•(T 3;Q) which is fixedunder the group G is in degree 0. �

The situation with the lower-dimensional strata is similar:

Lemma 5.3. Let σ(l) be an l-dimensional subcone of α1 ∗ α2 ∗ α3 ∗ γ1 ∗ γ2 ∗ γ3,with l ≥ 4. Then the stratum of γ3 associated to σ(l) has non-trivial cohomologywith compact support only in the maximal degree 2(6− l).

Proof. Recall that all GL(3,Z)-orbits of σ(l) were described in Lemma 5.1.Hence it suffices to consider the cases in which σ(l) is one of the following cones:

σ(4)I , σ

(4)II , σ

(5) := α1 ∗ α2 ∗ α3 ∗ γ1 ∗ γ2 and σ(6).

As mentioned above, if σ(l) is an l-dimensional cone in Σ then we have Tσ(l) =Cl × (C∗)6−l, because the fan Σ is basic. The corresponding stratum in AVor

3 isthen a quotient of {(0, 0, 0)} × (C∗)6−l ∼= (C∗)6−l by a finite group Gσ(l) . Toprove the claim, it suffices to show that the part of the cohomology of (C∗)6−l

which is invariant for the action of G coincides with H0((C∗)6−l;Q). Since (C∗)6−l

is smooth, the result about cohomology with compact support will follow fromPoincare duality.

For instance, consider the case of σ(4)II . Using toric coordinates, one finds that

the corresponding stratum is given by a quotient of (C∗)2 by the action of the finitegroup Z/2Z×S3. The factor S3 acts on γ1 ∗ α2 ∗ α3 by permuting γ1, α2 and α3,whereas the action of the factor Z/2Z is generated by the involution x1 ↔ −x1. Onecan compute explicitly the action of Z/2Z×S3 and prove (H•((C∗)2;Q))Z/2Z×S3 =H0((C∗)2;Q).

Analogous considerations yield the claim in the case of the other strata. �Concluding, the proof of Proposition 2.4 now follows from Lemmas 5.2 and 5.3.

References

[A] V. Alexeev, Complete moduli in the presence of semiabelian group action. Ann. of Math.(2) 155 (2002), 611–708.

[AN] V. Alexeev, I. Nakamura, On Mumford’s construction of degenerating abelian varieties,Tohoku Math. J. (2) 51 (1999), 399–420.

[Ar] D. Arapura, The Leray spectral sequence is motivic, Invent. Math. 160 (2005), no. 3,567–589.

[E] C. Erdenberger, A finiteness result for Siegel modular threefolds, Ph. D. Thesis, LeibnizUniversitat Hannover (2007). Available at http://www.iag.uni-hannover.de/∼ag-iag/

data/phdthesis erdenberger.pdf

[vdG] G. van der Geer, The Chow ring of the moduli space of abelian threefolds, J. AlgebraicGeom. 7 (1998), 753–770.

[G1] E. Getzler, Intersection theory on M1,4 and elliptic Gromov–Witten invariants, J. Amer.Math. Soc. 10 (1997), no. 4, 973–998.

[G2] E. Getzler, Topological recursion relations in genus 2, in: Integrable systems and algebraicgeometry (Kobe/Kyoto, 1997), World Sci. Publ., River Edge, NJ, 1998, 73–106.

[H] R. Hain, The rational cohomology ring of the moduli space of abelian 3-folds, Math. Res.Lett. 9 (2002), no. 4, 473–491.

[HKW] K. Hulek, C. Kahn, S. H. Weintraub, Moduli spaces of abelian surfaces: compactification,degenerations, and theta functions, de Gruyter Expositions in Mathematics 12. Walterde Gruyter & Co., Berlin, 1993.

[HS] K. Hulek, G. K. Sankaran, The geometry of Siegel modular varieties, Higher dimensionalbirational geometry (Kyoto, 1997), Adv. Stud. Pure Math., 35, 89–156, Math. Soc. Japan,Tokyo, 2002.

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[M1] D. Mumford, On the Kodaira dimension of the Siegel modular variety. In: Algebraicgeometry – open problems, proceedings, Ravello 1982, eds. C. Ciliberto, F. Ghione andF. Orecchia. Lecture Notes in Mathematics 997, Springer-Verlag, Berlin-New York, 1983,348–375.

[M2] D. Mumford, Towards an enumerative geometry of the moduli space of curves. In: Arith-metic and geometry, Vol. II. Progr. Math., 36, Birkhauser Boston, Boston, MA, 1983,271–328.

[O] M. Olsson, Compactifying moduli spaces for abelian varieties, Lecture Notes in Mathe-matics, 1958. Springer-Verlag, Berlin, 2008.

[PS] C. A. M. Peters, J. H. M. Steenbrink, Mixed Hodge Structures, Ergebnisse der Mathematikund ihrer Grenzgebiete. 3. Folge, 52, Springer-Verlag, Berlin, 2008.

[T] O. Tommasi, Geometry of discriminants and cohomology of modulispaces, Ph.D. thesis, Radboud University Nijmegen (2005). Available athttp://webdoc.ubn.ru.nl/mono/t/tommasi o/geomofdia.pdf

[Ts] R. Tsushima, A formula for the dimension of spaces of Siegel cusp forms of degree three,Amer. J. Math. 102 (1980), no. 5, 937–977.

[V] G. F. Voronoi, Nouvelles applications des parametres continus a la theorie des formesquadratiques. Deuxieme memoire. Recherches sur les paralleloedres primitifs. Secondepartie. Domaines de formes quadratiques correspondant aux differents types de par-alleloedres primitifs, J. Reine Angew. Math. 136 (1909), 67–178.

Leibniz Universitat Hannover, Institut fur Algebraische Geometrie, Welfengar-

ten 1, D-30167 Hannover, Germany


Leibniz Universitat Hannover, Institut fur Algebraische Geometrie, Welfengar-

ten 1, D-30167 Hannover, Germany


103


Quasi-Complete Homogeneous Contact ManifoldAssociated to a Cubic Form

Jun-Muk Hwang and Laurent Manivel

Dedicated to S. Ramanan

Abstract. Starting from a cubic form, we give a general construction of aquasi-complete homogeneous manifold endowed with a natural contact struc-ture. We show that it can be compactified into a projective contact manifold ifand only if the cubic form is the determinant of a simple cubic Jordan algebra.

1. Introduction

This note is at the crossroad of two different lines of study.On the one hand, we propose a general construction of a homogeneous quasi-

projective manifold Xc associated to a cubic form with a mild genericity property.These manifolds are rationally chain connected (Proposition 2.2), a property whichrelates our study to that of certain types of homogeneous spaces considered in[2, 3, 4].

On the other hand, we show that our manifolds Xc are endowed with naturalcontact structures (Proposition 3.1). Our construction thus appears as part of thegeneral study of contact projective and quasi-projective manifolds. Of course theprojective case is the most interesting one, the main open problem in this areabeing the Lebrun-Salamon conjecture: the only Fano contact manifolds should bethe projectivizations of the minimal nilpotent orbits in the simple Lie algebras. Asexplained in section 4, our construction is in fact modeled on these homogeneouscontact manifolds, which are known to be associated to very special cubic forms:the determinants of the simple cubic Jordan algebras.

Under both points of view, one of the most interesting questions one may askabout the quasi-projective contact manifolds Xc is about their compactifications.Even the existence of a small compactification (that is, with a boundary of codi-mension at least two) is not clear. Also, it is extremely tempting to try to constructnew projective contact manifolds by compactifying some Xc in such a way that thecontact structure extends. We show that this is possible if and only if the cubic c

2010 Mathematics Subject Classification. 14M99, 14M20, 14M17.The first author was supported by the Korea Research Foundation Grant funded by the

Korean Government (MEST)(KRF-2006-341-C00004).


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105

2 JUN-MUK HWANG AND LAURENT MANIVEL

is the determinant of a simple cubic Jordan algebra (Proposition 5.2). This can beinterpreted as an evidence for the Lebrun-Salamon conjecture.

2. Homogeneous spaces defined from cubics

Let V be a complex vector space of dimension p. Let c ∈ S3V ∗ be a cubic formon V . Let B : S2V → V ∗ be the system of quadrics defined by

B(v1, v2) = c(v1, v2, ·).In all the sequel we make the following

Assumption on c. The homomorphism B is surjective.

Let W be a complex vector space of dimension 2. Fix a choice of a non-zero2-form ω ∈ ∧2W ∗.

Let n := n1 ⊕ n2 ⊕ n3 where

n1 := V ⊗W, n2 := V ∗, n3 := W.

Define a graded Lie algebra structure on n, by

[v1 ⊗ w1, v2 ⊗ w2] = ω(w1, w2)B(v1, v2),

[v∗1 , v2 ⊗ w2] = v∗1(v2) w2.

The Jacobi identity holds because dimW = 2.Let N be the nilpotent Lie group with Lie algebra n. For a point � ∈ PW ,

denote by � ⊂ W the corresponding 1-dimensional subspace. Let

a� := V ⊗ � ⊂ V ⊗W = n1

be the abelian subalgebra of n and A� ⊂ N be the corresponding additive abeliansubgroup. We have the smooth subvariety A ⊂ N × PW defined by

A := {(g, �), g ∈ A�}.This variety A can be viewed as a family of abelian subgroups parametrized byPW . Let ψ : Xc → PW be the family of relative quotients

Xc := {N/A�, � ∈ PW}with the quotient map ξ : N × PW → Xc. Then

dimXc = dimN + 1− p = 2p+ 3.

Observe that ψ is a locally trivial fibration whose fibers are isomorphic to affinespaces. But it is not a vector nor an affine bundle. In fact the transition functionsare quadratic, because the nilpotence index of N is three.

The variety Xc is homogeneous under the action of the group

G := N � SL(W ) (semi-direct product).

Let o ∈ N be the identity and � ∈ PW be a fixed base point. Then x� := ξ(o× �)will be our base point for Xc. Its stabilizer is H = A� � B�, if B� denotes thestabilizer of � in SL(W ). Moreover a Borel subgroup of G is B = N � B�, and wehave a sequence of quotients

G→G/B�ξ→ G/H = Xc

ψ→ G/B = PW.

Proposition 2.1. (1) Xc is simply connected.(2) Let L := ψ∗OPW (1). Then Pic(Xc) = ZL.

106

HOMOGENEOUS CONTACT MANIFOLD 3

Proof. As a variety, each A� is nothing but an affine space. So the varietyXc being fibered in simply connected manifolds over the projective line, is simplyconnected. This proves (1).

The character group X(G) of G being trivial, the forgetful map

α : PicG(Xc)→Pic(Xc)

is injective ([9] Proposition 1.4). Moreover the Picard group of G is trivial, so α is infact an isomorphism (see the proof of Proposition 1.5 in [9]). But PicG(Xc) � X(H)and an easy computation shows that X(H) = X(B�). This implies (2). �

Note that G is generated by H and SL(W ) such that H ∩ SL(W ) is a Borelsubgroup of SL(W ). Thus we can apply Proposition 4.1 in [4] to deduce:

Proposition 2.2. The variety Xc is rationally chain connected. In particular,Xc is quasi-complete, i.e., there is no non-constant regular function on Xc.

In fact, it is easy to show that for any n ∈ N , the image of {n} × PW under ξis a smooth rational curve on Xc with normal bundle of the form O(1)p ⊕ Op+2.

3. Contact structures

Consider the tangent spaces

To×�(N × PW ) = n1 ⊕ n2 ⊕ n3 ⊕ T�(PW )

Tx�(Xc) = n1/a� ⊕ n2 ⊕ n3 ⊕ T�(PW ).

Using the subspace � ⊂ W = n3, we define the hyperplane

Dx�:= n1/a� ⊕ n2 ⊕ �⊕ T�(PW )

inside Tx�(Xc). This hyperplane is invariant under the action of the stabilizer H of

x� in G = N �SL(W ), so we get a well-defined hyperplane distribution D ⊂ T (Xc)with T (Xc)/D ∼= L.

Proposition 3.1. The distribution D ⊂ T (Xc) defines a contact structure.

Remark. Observe that since

Dx�= (V ⊗W/�) ⊕ V ∗ ⊕ �⊕Hom(�,W/�),

there is a natural W/�-valued symplectic pairing on Dx�. Note that W/� is the

fiber of L at x�. This shows that the bundle D has an L-valued symplectic form,and indeed this symplectic form comes from the contact structure

0 → D → T (Xc) → L → 0.

Proof. Let Yc be the variety defined as the complement L× of the zero sectionin the total space of the line bundle dual to L. Let θ be the L-valued 1-form on Xc

defining D. To check that θ is a contact form, it suffices to show that the 2-formdθ where θ is the pull-back of θ to Yc, is symplectic (see [1], Lemma 1.4).

To check this we make a local computation. Letm ∈ PW be some point distinctfrom � and m ⊂ W be the corresponding line. Then

nm := (V⊗m) ⊕V ∗⊗W ⊂ n

defines a complement to a� ⊂ n. We can define a local analytic chart on Xc aroundx� by

x(X, p) = ξ(exp(X)× p),

107


where X ∈ nm and p ∈ PW − m. Let us write down θ in that local chart. Sincethe chart preserves the fibration over PW we just need to compute over �. Thedifferential eX of the exponential map at X, seen as an endomorphism of nm, isdefined by the relation

exp(X + teX(Y ) +O(t2))x� = exp(X)exp(tY )x�.

Now we can use the fact that N being 3-nilpotent, the Campbell-Hausdorff formulain N is quite simple: we have exp(X)exp(Y ) = exp(H(X,Y )) for X,Y ∈ n, with

H(X,Y ) = X + Y +1

2[X,Y ] +

1

12[X, [X,Y ]] +

1

12[Y, [Y,X]].

We easily deduce that eX(Y ) = Y + 12 [X,Y ]+ 1

12 [X, [X,Y ]]. Now we can decomposethis formula with respect to the three-step grading of n. If Z = eX(Y ) = Z1+Z2+Z3, we find that

Z1 = Y1,

Z2 = Y2 +1

2[X1, Y1],

Z3 = Y3 +1

2[X1, Y2] +

1

2[X2, Y1] +

1

12[X1, [X1, Y1]],

which can be inverted as

Y1 = Z1,

Y2 = Z2 −1

2[X1, Z1],

Y3 = Z3 −1

2[X1, Z2]−

1

2[X2, Z1] +

5

12[X1, [X1, Z1]].

Since the hyperplane Dx�is defined by the condition that Y3 belongs to �, we deduce

that the contact form is given at x(X, �), in our specific chart, by the formula

θx(X,�)(Z) = Z3 −1

2[X1, Z2]−

1

2[X2, Z1] +

5

12[X1, [X1, Z1]] mod �.

Even more explicitly, if we write Z1 = z1⊗m andX1 = x1⊗m, we have [X1, Z1] = 0,[X1, Z2] = Z2(x1)m and [X2, Z1] = −X2(z1)m, so

θx(X,�)(Z) = Z3 +1

2(X2(z1)− Z2(x1))m.

Now we pull-back θ to Yc = L×. A local section of L× around � is given by m∗−z�∗

over the point p = �+ zm of PW . Over φ = y(m∗ − z�∗), we get the 1-form on L×

given in our local chart by

θx(X,p),φ(Z, Y ) = y(m∗ − z�∗)(Z3) +y

2(X2(z1)− Z2(x1)).

If Z3 = Z13m+ Z2

3 �, this can also be written as:

θx(X,p),φ = y(dX13 − zdX2

3 ) +y

2(X2dX1 −X1dX2).

We can easily differentiate this expression and evaluate it at x�. We obtain

dθx�,ym∗ = dy ∧ dX13 − ydz ∧ dX2

3 + ydX2 ∧ dX1.

Since y is a non zero scalar this 2-form is everywhere non-degenerate. By homo-geneity this remains true over the whole of L×, and the proof is complete. �

108


4. Projective homogeneous contact varieties

Consider a complex simple Lie algebra g and the adjoint variety

Yg = POmin ⊂ Pg,

the projectivization of the minimal nilpotent orbit Omin. Then Yg is homogeneousunder the action of the adjoint group G = Aut(g). Suppose that Pic(Yg) � Z

(this is the case if and only if g is not if type A). Then the variety F of lines onYg is G-homogeneous and we can describe a line as follows. Choose T ⊂ B ⊂ G amaximal torus and a Borel sugbroup. Let gψ denote the root space in g associatedto the highest root ψ. Then Yg = Ggψ and the stabilizer of gψ is the maximalparabolic subgroup Pα of G defined by the unique simple root α such that ψ − αis a root. Moreover the line � = 〈gψ, gψ−α〉 is contained in the adjoint variety Yg,and F = G.�.

There is a five-step grading on g defined by the highest root ψ, as follows.Define Hψ ∈ [gψ, g−ψ] by the condition that ψ(Hψ) = 2. Then the eigenvalues ofad(Hψ) are 0,±1,±2 and the eigenspace decomposition yields the five-step grading

g = g−2 ⊕ g−1 ⊕ g0 ⊕ g1 ⊕ g2.

We have g2 = gψ, while gα and gψ−α are respectively lines of lowest and highestweights in g1.

Since gψ−α defines, exactly as gψ, a point of the adjoint variety, we can usethe root ψ − α to define another five-step grading. Since ad(Hψ) and ad(Hψ−α)commute, we get a double grading on g. Moreover, the stabilizer s ⊂ g of the line� ⊂ g decomposes as follows (where the grading defined by ad(Hψ) can be readhorizontally):

g−α

g00 g10

gα g11 g21 gψ−α

gψ

Let W = gψ−α ⊕ gψ � C2 and V = g∗21. The map g10 ⊗ g21 → g31 = gψ−α definedby the Lie bracket is a perfect pairing, as well as g11 ⊗ g21 → g32 = gψ, giving anatural identification

g10 ⊕ g11 � V⊗W

and isomorphisms

g10φ−→ g11

∼= V.

The positive part of the vertical grading of s thus reads

(V⊗W ) ⊕V ∗⊕W = n.

Note that the degree zero part of this grading reads sl(W ) × h00, where g00 =[g−α, gα]⊕h00 is an orthogonal decomposition with respect to the Killing form.

The cubic form c on V is defined (up to scalar) once we identify g10 with g11,through the map φ. We also need to choose a generator Xψ of gψ. Then we candefine c by the formula

[φ(X), [φ(X), X]] = c(X)Xψ ∀X ∈ g10 � V.

Remark. This construction is closely related to the ternary models for simple Liealgebras considered in [7], section 2. These models are of the form

g = h× sl(U) ⊕ (U⊗V ) ⊕ (U∗⊗V ∗),

109


where U is three dimensional, and V is an h-module. To define a Lie bracket on g,one needs a cubic form c on V , a cubic form c∗ on V ∗, and a map θ : V⊗V ∗→h.Then the Jacobi identity implies a series of conditions on these data, including that

h ⊂ Aut(c) ∩Aut(c∗).

These conditions should ultimately lead to a cubic Jordan algebra structure on V .If we choose a maximal torus in sl(U) and use the associated grading on U,U∗, weget an hexagonal model as in Figure 2 of [8] :

g−α

g−ψ g−1−1 g10 gψ−α

g−2−1 g00 g21

gα−ψ g−10 g11 gψ

gα

The subalgebra we denoted n is the sum of the factors in the last three columns.and we can add the factor sl(W ) from the middle column in order to get s.

Once we have defined the cubic c associated to the simple Lie algebra g, wehave the associated homogeneous space Xc with its natural contact structure. Adirect verification gives:

Proposition 4.1. The homogeneous space Xc is an open subset of Yg, witha codimension two boundary. Its contact structure is the restriction of the naturalcontact structure on Yg.

It is tempting, but illusory, as we shall see, to try to construct new projectivecontact manifolds as suitable compactifications of our homogeneous spaces Xc forother types of cubics.

5. Compactifications

Since all regular functions on Xc are constant, we can expect that Xc admitsa small compactification, that is, a projective variety Xc containing Xc as an opensubset in which the boundary of Xc has codimension two or more. By Theorem 1in [3] and Proposition 2.1, it is enough to check that the algebra of sections

R(Xc, L) =∞⊕k=0

Γ(Xc, Lk)

is of finite type, as well as all the R(Xc, Lm), for m ≥ 1. We have not been able to

prove this but we can make the following observations.

Since the Lie algebra g of G = N � SL(W ) preserves the contact structure wehave defined on Xc, there must be a morphism ϕ from Xc into Pg∗ (see [1], Section1). In fact, any contact vector field on a contact manifold defines a holomorphicsection of the contact line bundle L. Thus g defines a linear subsystem in |L|. Themorphism is always etale over its image, and since our Xc is simply connected weconclude that ϕ embeds Xc as a coadjoint orbit in g∗. We thus have a naturalprojective compactification of Xc in Pg∗.

110


Note that the inclusion of Xc in Pg∗ is just the projectivization of the momentmap of the symplectic variety Yc = L×. We have a commutative diagram

Ycμ−→ g∗

↓ ↓Xc

ν−→ Pg∗

Here μ denotes the (G-equivariant) moment map and ν is its quotient by the C∗-

action. We have g∗ = n∗ ⊕ sl(W )∗ and the component μ′ of μ on n∗ is not injective,since the N action on Xc preserves the P1-fibration. Consider μ′(Yc) ⊂ n∗.

Proposition 5.1. Suppose that the cubic hypersurface Zc ⊂ PV be smooth.Then the boundary of μ′(Yc) has codimension at least two.

Proof. We can describe explicitly the closure of μ′(Yc) as the set of triples(φ1, φ2, φ3) ∈ n∗ such that

ω(φ1, φ3) = c(φ2, φ2, .),

where ω : (V ∗⊗W )×W → V ∗ is the natural bilinear map.If φ3 �= 0, we are in μ′(Yc). Thus on the boundary, we must have φ3 = 0, and

then c(φ2, φ2, .) = 0. But under our smoothness assumption on Zc, this impliesthat φ2 = 0. So the boundary of μ′(Yc) has dimension at most 2p = dimμ′(Yc)−2,the number of parameters for φ1. �

This seems to be a first step towards proving that Xc has a small compactifi-cation. But we have not been able even to find conditions on c that would ensurethat the compactification Xc ⊂ Pg∗ is small.

What is rather surprising is that the cubics whose associated variety Xc has asmooth contact compactification can be completely classified. By this, we mean asmooth projective variety Xc compactifying Xc, with a contact structure extendingthat of Xc.

Proposition 5.2. There exists a smooth contact compactification Xc of Xc ifand only if c is the cubic norm of a semi-simple Jordan algebra.

Proof. We will deduce this statement from a study the variety of minimalrational tangents Cx�

⊂ PDx�. Note that the space of lines on Xc through x� is just

ξ−1(x�) ∼= A�∼= V .

We claim that the tangent map sending a line through x� to its tangent directionin PDx�

is equal, up to scalars, to the rational map

τ : V → PDx�= P(V ⊕ V ∗ ⊕ C⊕ C)

τ (v) := [v : B(v, v) : c(v, v, v) : 1].

Indeed, a line through x� in Xc is of the form �g = ξ(g× PW ) for g ∈ A�. To writethis line in the local chart we used in the proof of Proposition 3.1 (we use the samenotations), we must write

ξ(g × p) = exp(Z)ξ(o× p).

If g = exp(X) with X ∈ V⊗�, this amounts to solving the equation exp(X) =exp(Z)exp(W ), with Z ∈ nm and W ∈ V⊗p. So X = H(Z,W ), and if we write

111


X = v⊗� for some v ∈ V , we must have W = v⊗p and then we get, up to term oforder at least two in z,

Z1 = −zv⊗m,

Z2 = −1

2[Z1,W ] =

z

2B(v, v),

Z3 = −1

6[Z1, [Z1,W ]] = −z

6c(v)�.

This proves the claim.We can now conclude the proof as follows. By the results of [5], the closure of

the image of this map must be smooth if there exists a smooth contact compactifi-cation of Xc. So the closure of the image of the map τ must be smooth. But thenwe can apply Corollary 26 in [6]. �

References

[1] Beauville A., Fano contact manifolds and nilpotent orbits, Comment. Math. Helv. 73 (1998),no. 4, 566–583.

[2] Bien F., Borel A., Sous-groupes epimorphiques de groupes lineaires algebriques I, C. R. Acad.Sci. Paris Ser. I Math. 315 (1992), no. 6, 649–653.

[3] Bien F., Borel A., Sous-groupes epimorphiques de groupes lineaires algebriques II, C. R.Acad. Sci. Paris Ser. I Math. 315 (1992), no. 13, 1341–1346.

[4] Bien F., Borel A., Kollar J., Rationally connected homogeneous spaces, Invent. Math. 124(1996), no. 1-3, 103–127.

[5] Kebekus S., Lines on complex contact manifolds II, Compos. Math. 141 (2005), no. 1, 227–252.

[6] Landsberg J.M., Manivel L., Legendrian varieties, Asian J. Math. 11 (2007), no. 3, 341–359.[7] Manivel L., Configurations of lines and models of Lie algebras, J. Algebra 304 (2006), no. 1,

457–486.[8] Mukai S., Simple Lie algebra and Legendre variety, preprint 1998.[9] Mumford D., Fogarty J., Kirwan F., Geometric invariant theory, Third edition, Springer

1994.

Jun-Muk Hwang: Korea Institute for Advanced Study, Hoegiro 87, Seoul, 130-722,

Korea


Laurent Manivel: Institut Fourier, UMR 5582 CNRS/UJF, Universite Joseph Fou-

rier, 38402 Saint Martin d’Heres, France


112


Maximal Weights in Kahler Geometry: Flag Manifolds andTits Distance (with an Appendix by A. W.

I. Mundet i Riera

Dedicated to Professor Ramanan, on the occasion of his 70th birthday

Abstract. We review the definition of maximal weights and polystability inKaehler geometry given in [M]. We compute the maximal weights in flag vari-eties, and we relate them to the Tits distance between points of the boundaryat infinity of the corresponding symmetric space. In the appendix by A.H.W.Schmitt a proof is given of an algebraic analogue of the characterization ofpolystability in terms of maximal weights.

1. Introduction

The purpose of this paper is twofold. First, we survey the main constructionsand results in [M], notably

• the maximal weight functions on the boundary at infinity ∂∞(K\KC)associated to a Hamiltonian action of a compact Lie group K on a Kahlermanifold X extending to a holomorphic action of KC, and

• the characterization of points in X whose KC-orbit intersects the zerolocus of the moment map in terms of maximal weights.

Second, we compute the maximal weights in some particular examples. The onlynew result in this paper is a formula relating the maximal weights for the standardactions on flag manifolds to the Tits distance between points in the boundaryat infinity of symmetric spaces of noncompact type. This formula is analogousto formula (10) in [KLM], but here we adopt a different point of view, puttingmore emphasis on the symplectic aspects of the formula. Compared to [KLM],our perspective is wider in the sense that we consider actions on arbitrary Kahlermanifolds (the results in [KLM] are related to actions on flag manifolds), butit is more restrictive in the sense that we only consider symmetric spaces of theform K\KC ([KLM] considers arbitrary symmetric spaces of noncompact type).However, it is very likely that the construction given in [M] extends to the settingconsidered in [HS], and such an extension should make use of arbitrary symmetricspaces of noncompact type.

2010 Mathematics Subject Classification. Primary 53D20; Secondary 32M05.


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113

H Schmitt).

2 I. MUNDET I RIERA (APPENDIX BY A.H.W. SCHMITT)

1.1. Let us recall the setting of [M]. Denote by X a connected Kahler manifoldendowed with an action by holomorphic isometries of a compact Lie group K.Assume that this action extends to a holomorphic action of the complexificationof K, which we denote by G, and that the action of K on X is Hamiltonian withrespect to the symplectic form ω defined as the imaginary part of the Hermitianpairing in X. Denote by k the Lie algebra of K. The action of K on X beingHamiltonian, there exists a K-equivariant map (the so-called moment map)

μ : X → k∗

such that, for any s ∈ k, we have 〈dμ, s〉 = ιXsω, where Xs is the vector field on

X generated by the infinitesimal action of s and ιXsω is the contraction of ω by

Xs. We assume that the length of the vector Xs(x) grows at most linearly as afunction of the distance of x to some base point x0 ∈ X. More precisely, we makethe following

Assumption 1.1. There exists a constant C > 0 such that for any x ∈ X andany s ∈ k we have

|Xs(x)| ≤ C |s| (1 + dX(x0, x)),

where dX denotes the geodesic distance between points of X.

Two natural situations in which this assumption holds are the case of compactX and that in which X is a Hermitian vector space with an action of K by unitarytransformations.

Let ∂∞(K\G) denote the boundary at infinity of the symmetric space K\G(see Section 2.1 for a reminder of its definition). In [M] we defined, for each pointx ∈ X, a map

λx = λ(X,ω)x : ∂∞(K\G) → R ∪ {∞},

and we defined x to be analytically polystable if and only if:

(1) for any y ∈ ∂∞(K\G) we have λx(y) ≥ 0,(2) if y ∈ ∂∞(K\G) satisfies λx(y) = 0, then there exists another point y′ ∈

∂∞(K\G) satisfying λx(y′) = 0 and there is a geodesic in K\G which

converges on one side to y and on the other side to y′.

The main result in [M] is:

Theorem 1.2. x is analytically polystable if and only if μ−1(0) ∩ G · x isnonempty. Furthermore, if μ−1(0) ∩G · x is nonempty then it consists of a uniqueK-orbit.

The interest of such a result stems from the fact that the quotient μ−1(0)/Kis in a natural way a singular symplectic manifold (usually called the symplectic orMarsden–Weinstein quotient), which thanks to the theorem can be identified withthe quotient of the set Xpol = {x ∈ X | x is analytically polystable} by G.

It clearly follows from Theorem 1.2 that x is polystable if and only if any pointin its G-orbit is polystable. Such a statement is also a consequence of the followingequivariance property of maximal weights, which was proved in [M] (see Lemma2.1 below for a sketch of the proof).

Lemma 1.3. For any x ∈ X, g ∈ G and y ∈ ∂∞(K\G) we have λg·x(y) =λx(y · g).

In the proof of this lemma we crucially use Assumption 1.1.

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MAXIMAL WEIGHTS IN KAHLER GEOMETRY AND TITS DISTANCE 3

1.2. Suppose that X is a projective variety, that the action of G on X lifts toa linear action on a very ample line bundle L on X, and that the Kahler form on Xis the restriction of the Fubini–Study form of the projective embedding given by L.In this situation a classical theorem of Kempf and Ness states that for any x ∈ X

(1.1) G · x ∩ μ−1(0) �= ∅ ⇐⇒ G · lx is closed ,

where lx is any nonzero element in the fiber of L → X over x (see Theorem 4.11 in[T]).

If the condition on the right hand side in (1.1) holds, then x is said to bepolystable. So putting together Theorem 1.2 and (1.1) we deduce that in the pro-jective situation analytic polystability is equivalent to polystability. Furthermore,in the projective situation the maximal weight function can be described in purelyalgebraic terms: up to some constant factors, it coincides with the maximal weightsdefined in Geometric Invariant Theory (GIT) [MFK] (see Section 2.5). It followsthat there is an algebraic characterization, in terms of GIT maximal weights, ofpolystable points in X. This characterization, which extends the classical Hilbert–Mumford criterion, seems to be new in GIT, at least as an explicit statement. Thisis not so surprising: from the point of view of algebraic geometry it is not as naturalto consider the set of polystable points Xpol ⊂ X as it is to consider the sets ofstable and semistable points, since unlike the latter Xpol is not Zariski open. Butof course the condition of a point being polystable appears recurrently in problemsof GIT, as for example in the construction of moduli spaces of principal sheaves(see for example [S]).

Shortly after the appearance of [M], A. Schmitt provided a completely algebraicproof of the Hilbert–Mumford criterion for polystability, which is included as anappendix of the present paper.

1.3. We now describe our formula for the maximal weights of flag manifolds.Recall that a flag manifold of G is a homogeneous space of the form X = G/P ,where P is a parabolic subgroup of G. Fix one such X. It is clear that X carriesa G-invariant complex structure. In order to apply the previous constructions toX we need to endow it with a K-invariant Kahler structure admitting a momentmap μ : X → k∗. This is exactly the same thing as choosing a K-equivariantdiffeomorphism of X with a coadjoint orbit O∗ ⊂ k∗ (see Section 3.2 for details).Assume that this choice has been made, and denote by ω the resulting invariantsymplectic form on X. Since X is compact, Assumption 1.1 holds, so we can indeedapply toX the previous definitions, and maximal weights satisfy the G-equivarianceproperty stated in Lemma 1.3.

Fixing an invariant Euclidean norm on k we obtain:

• a G-invariant Riemannian metric on K\G, which can be used to definethe Tits distance on ∂∞(K\G) (see Section 3.1), and

• a K-equivariant isomorphism k∗ � k, which identifies the coadjoint orbitO∗ � X with an adjoint orbit O ⊂ k.

Assume that the orbit O is contained in the sphere S(k) ⊂ k of vectors of norm 1 (thiscan always be assumed if we conveniently rescale the symplectic form on X; on theother hand, rescaling the symplectic form has the effect of multiplying the maximalweight function by the same factor, so it is easy to modify the formula in Theorem1.4 so that the hypothesis that O ⊂ S(k) is unnecessary). Let e : O → ∂∞(K\G)

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be the map which sends s ∈ O to the class of the geodesic ray t �→ [eits]. Let

j : X → ∂∞(K\G)

be the composition of the isomorphism X → O with the map s. In Section 3.3 weprove:

Theorem 1.4. For any x ∈ X and any y ∈ ∂∞(K\G) we have

λ(X,ω)x (y) = − cos dT (j(x), y),

where dT denotes the Tits distance.

The idea that Tits distance is related to stability notions on homogeneousspaces appears in the work of M. Kapovich, B. Leeb and J. Millson [KLM]. Inparticular, Theorem 1.4 can be seen, adopting a suitable point of view, as a restate-ment of formula (10) in [KLM]. We discuss in some detail the relation betweenTheorem 1.4 and the results in [KLM] in Section 3.4 below.

We remark that in the proof of Theorem 1.4 we only use the definition ofthe Tits distance combined with the equivariance property of the maximal weightfunction stated in Lemma 1.3. As a consequence, we obtain a new proof of thefollowing well known fact (see Section 2.1 for some explanations of the notionsappearing in the statement).

Corollary 1.5. Let y, y′ ∈ ∂∞(K\G) be two arbitrary points. Let ξ : [0,∞) →K\G be a geodesic ray representing y. For any t ∈ [0,∞) there exists a uniquegeodesic ray ηt : [0,∞) satisfying ηt(0) = ξ(t) and representing y′. We then have

dT (y, y′) = lim

t→∞Angle(ξ′(t), η′t(0)),

where the angle takes values in [0, π].

Another consequence of Theorem 1.4 is a certain symmetry property of maximalweights of flag manifolds. To state this property we introduce some notation. Givenany y ∈ ∂∞(K\G) we denote by Fy the orbit G ·y and by ωy the natural symplecticform on Fy. Combining Theorem 1.4 with the symmetry of Tits distance, wededuce:

Corollary 1.6. For any y, y′ ∈ ∂∞(K\G) we have

λ(Fy,ωy)y (y′) = λ

(Fy′ ,ωy′ )

y′ (y).

Acknowledgements. We thank the referee for a very careful reading of the paper,and for pointing out a mistake and a number of typos in an earlier version of thispaper.

2. Maximal weights and polystability

In this section we recall the main constructions in [M] and we compute someexamples. For the reader’s convenience we have expanded some of the explanationsin [M], especially those recalling the (well known among experts) construction ofthe boundary at infinity of K\G and the action of G on it; however, the proofs ofthe new results in [M] are only sketched, so the reader might wish to consult [M]for more details.

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2.1. The boundary ∂∞(K\G). Let K be a compact Lie group, and let Gbe its complexification. Before defining the boundary ∂∞(K\G) we first recallthe definition of the symmetric space K\G. The coset space K\G has a naturalstructure of smooth manifold with respect to which the action of G on the right issmooth. The stabilizer of the class of the identity [1] ∈ K\G is K, so there is anatural linear action of K on T[1](K\G). Furthermore we can identify T[1]K\G �k\g. Using the splitting g = k⊕ ik we obtain an isomorphism T[1](K\G) � ik, whichis K-equivariant when we consider on ik � k the adjoint action of K. Consequently,any Euclidean norm on k invariant under the adjoint action induces a G-invariantRiemannian metric on K\G. Assume from now on that such a structure has beenchosen. It follows that the action of G on K\G is by isometries. Although there aredifferent possible choices of G-invariant Riemannian metrics on K\G, the followingproperties always hold, regardless of the choice:

(1) any geodesic in K\G can be parameterized by a map t �→ [eitsg] ∈ K\G,where s ∈ k and g ∈ G,

(2) the curvature of K\G is everywhere nonpositive (see [E] for a computa-tion),

(3) K\G is a symmetric space: this means that for any x ∈ K\G there isan isometry which fixes x and which acts on the tangent space at x asmultiplication by −1.

It is a very natural and classical problem to compactify the symmetric spaceK\G (which, as a manifold, is diffeomorphic to an Euclidean space) in such a waythat the action of G on K\G extends to the compactification. One can compact-ify symmetric spaces in many different ways, and which compactification is mostadequate depends on one’s particular needs (a description of many of the possiblecompactifications can be found for example in [BJ]). The set of points which oneadds to compactify is called the boundary. The compactification which we will de-scribe here is homeomorphic to a closed ball and the boundary, which we denote by∂∞(K\G), is usually called the visual boundary and is homeomorphic to a sphere.

A geodesic ray is a map γ : [0,∞) → K\G giving a parametrization by arclength of a portion of geodesic. Let d denote the distance function between pointsin K\G. Two geodesic rays γ0, γ1 are declared to be equivalent, and denotedγ0 ∼ γ1, if the distance d(γ0(t), γ1(t)) is bounded independently of t. This definesan equivalence relation. As a set, the boundary at infinity ∂∞(K\G) is:

∂∞(K\G) = { geodesic rays }/ ∼ .

Since G acts on K\G by isometries, it acts on the set of geodesic rays preservingthe equivalence relation ∼, so there is a natural action of G on ∂∞(K\G).

Now we define a topology on ∂∞(K\G). Let S(k) ⊂ k denote the unit sphere.For any s ∈ S(k) we define es ∈ ∂∞(K\G) to be the class of the geodesic rayηs : (0,∞) → K\G defined as ηs(t) = [eits]. Then the map

e : S(k) � s �→ [es] ∈ ∂∞(K\G)

is a bijection (see Section II.2 in [B]) and we take on ∂∞(K\G) the unique topologywith respect to which e is a homeomorphism. Then the action of G on ∂∞(K\G)is by homeomorphisms.

For each s ∈ S(k) and any g ∈ G define s · g ∈ S(k) by the property that

es · g = es·g.

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This action of G on S(k) extends the antiadjoint action of K on S(k), defined ass · h = Ad(h−1)(s) for any s ∈ k and h ∈ K. However, unlike the action of K, theaction of G is in general not smooth.

We now describe the compactification obtained by adding toK\G the boundary

∂∞(K\G). Define K\G := (K\G)∪∂∞(K\G). There is a unique topology on K\Gwhich satisfies these two properties:

(1) the inclusions of K\G and ∂∞(K\G) in K\G give homeomorphisms withtheir image,

(2) if {xi} ⊂ K\G is a diverging sequence then one can write xi = ηsi(ti),where si ∈ S(k) and ti are real numbers converging to ∞; then xi → es ifand only if si → s.

It is straightforward to prove that this topology on K\G is compact.Since the parameterized geodesics do not depend on the choice of invariant

metric on k and the distance functions on K\G induced by two choices of invariantmetric on k are uniformly comparable, the boundary ∂∞(K\G) and the compact-

ification K\G are completely canonical as topological spaces endowed with a Gaction.

2.2. Example: compactifying U(n)\GL(n,C). It is instructive to studyin concrete terms the previous definitions in the case of the symmetric spaceU(n)\GL(n,C), which parameterizes Hermitian metrics on Cn.

Let s ∈ S(u(n)). The matrix is is Hermitian symmetric, so it diagonalizes andhas real eigenvalues, say λ1 < · · · < λr. Let Vj = Ker(λj − is) be the eigenspacecorresponding to λj and define V k = V1 ⊕ · · · ⊕ Vk for any integer k ≥ 1.

Now let g ∈ GL(n,C) be any element, and define

V ∞j = (g−1(V j−1))⊥ ∩ g−1(V j),

where V ⊥ denotes the orthogonal of V . Then we have a direct sum decompositionC

n =⊕

V ∞j . Define ρg(s) ∈ u(n) by the conditions that ρg(s) preserves each V ∞

j

and that the restriction of ρg(s) to V ∞j is given by multiplication by −iλj . We

claim that

(2.2) ρg(s) = s · g.We remark that if we prove the previous formula for any s and g then we obtain asa consequence the fact that the map

e : S(u(n)) → ∂∞(U(n)\GL(n,C))

is a bijection, since, as we have observed earlier, any geodesic ray in U(n)\GL(n,C)is of the form t �→ eitsg for some s ∈ u(n) and g ∈ GL(n,C).

We sketch two proofs of (2.2).First proof. We use the fact that the curvature of U(n)\GL(n,C) is nonpositive.This implies the following convexity property: if γj : [0, t] → U(n)\GL(n,C),j = 1, 2, are two geodesic segments parameterized at uniform speed, then for anyλ ∈ [0, t] we have

d(γ1(λ), γ2(λ)) ≤ max{d(γ1(0), γ2(0)), d(γ1(t), γ2(t))},where d is the distance function between points in U(n)\GL(n,C). Now define themap

log : GL(n,C) → u(n)

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by the condition that log(g) = u if g = keiu is the Cartan decomposition of g, sothat k ∈ U(n) and u ∈ u(n). Assume that the following limit exists:

u = limτ→∞

1

τlog(eiτsg).

We claim that then s · g = u. In fact, we have the following uniform upper bound:

(2.3) d([eiτu], [eiτsg]) ≤ d(1, [g]) for any τ ≥ 0.

To prove (2.3) define, for any t > 0, ut = t−1 log(eitsg), so that eitut = eitsg. Fixsome t > 0 and define the maps

γt1, γ2 : R → U(n)\GL(n,C), γt

1(η) = [eiηut ], γ2(η) = [eiηsg].

Both γ1 and γ2 parameterize geodesics at uniform speed, so the convexity propertyimplies, for any τ ≥ 0,

d(γt1(τ ), γ2(τ )) ≤ max{d(γt

1(0), γ2(0)), d(γt1(t), γ2(t))}

= max{d([1], [g]), d([eitut ], [eitsg])}= max{d([1], [g]), 0} = d([1], [g]).

Since by assumption ut → u, the continuity of the exponential map implies γt1(τ ) →

eiτu as t → ∞, which combined with the continuity of the distance function implies(2.3).

The conclusion is that (2.2) follows from this formula:

(2.4) ρg(s) = limτ→∞

1

τlog(eiτsg).

To prove (2.4) one can argue as follows. Take any ε > 0 smaller than inf{λj −λj−1}/3. Using the variational description of eigenvalues and eigenspaces of log(h),one proves that for big enough t the eigenvalues of ut are contained in

⋃[λj−ε, λj+

ε], and the number of eigenvalues in [λj − ε, λj + ε] is equal to dimVj . Furthermore,if we let V t

j be the direct sum of the eigenspaces of ut with eigenvalue contained in

[λj − ε, λj + ε], then V tj converges to V ∞

j in the Grassmannian variety.

Second proof. The second proof is more direct and elementary, since it does notuse any curvature properties, but it is perhaps not as enlightening as the first one.Define for convenience u := ρg(s) and Wi := V ∞

i . We first observe that

(2.5) d([eitsg], [eitu]) = d([eitsge−itu], [1]),

since GL(n,C) acts on the symmetric space by isometries. To give a uniform upperbound for the term on the right hand side of (2.5) is equivalent to giving someconstant C > 0 such that for any t ≥ 0 and any x ∈ Cn we have

(2.6) C−1|x|2 ≤ |eitsge−itux|2 ≤ C|x|2.

Using induction on j one proves that for any j we have g(W1 ⊕ · · · ⊕ Wj) = V j .Consequently, we can write g =

∑j≥k gjk where gjk is a map from Wj to Vk. In

particular each map gjj is invertible, because g is invertible. Now let x ∈ Cn beany element, and write x =

∑j xj with xj ∈ Wj . We compute:

eitsge−itux =∑j

gjj(xj) +∑j>k

gjk(xj)et(λk−λj).

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Since the vectors {gjj(xj)} are pairwise orthogonal and the matrices gjj are invert-ible, there is some constant C1 independent of x such that

C−11 |x|2 ≤ |

∑j

gjj(xj)|2 ≤ C1|x|2.

On the other hand, setting η = inf{λj −λj−1} there is a constant C2 > 0, indepen-dent of x, such that

|∑j>k

gjk(xj)et(λk−λj)|2 ≤ C2|x|2e−tη.

Since η > 0, combining this estimate with the previous one we obtain, for a suffi-ciently big C > 0 and any t, the inequality (2.6).

2.3. Maximal weights. We now come back to the general setting presentedin the introduction, so X is a connected Kahler manifold endowed with a Hamilton-ian action of K with moment map μ : X → k∗, and there is a holomorphic action ofG on X extending the action of K. Let x ∈ X be any point. The maximal weightfunction

λx : ∂∞(K\G) → R ∪ {∞}is defined as

λx(es) := limt→∞

〈μ(eits · x), s〉 ∈ R ∪ {∞}.

This limit exists because 〈μ(eits · x), s〉 is nondecreasing as a function of t, whichalso implies that

(2.7) λx(es) ≥ 〈μ(x), s〉.The most important property of the maximal weight function proved in [M] is thefollowing equivariance formula, which was stated in the introduction:

Lemma 2.1. For any x ∈ X, g ∈ G and y ∈ ∂∞(K\G) we have

(2.8) λg·x(y) = λx(y · g).

Proof. (Sketch.) Define a function Ψx : K\G → R as:

Ψx([eiu]) =

∫ 1

0

〈μ(eitu · x), u〉 dt.

We call Ψx the integral of the moment map. For any x ∈ X and g ∈ G we havedΨg·x = ρ∗gdΨx, where ρg : K\G → K\G is the map given by multiplication by g onthe right (to prove this formula one uses the integrability of the complex structureand of the symplectic form on X, so the condition that X is Kahler seems to becrucial for this result to hold). Integrating we deduce the following cocycle property

(2.9) Ψx([gh]) = Ψx([g]) + Ψg·x([h]).

Now the maximal weight function is related to the integral of the moment map asfollows:

λx(es) = limt→∞

1

tΨx(e

its · x).In fancy words, λx is a sort of renormalized limit of Ψx as one approaches the bound-ary of K\G. Combining this formula with the cocycle property (2.9) one deduces(2.8). This deduction, however, is not completely immediate, and some controlis needed on the growth of Ψx; in [M] we obtain such control using Assumption1.1. �

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2.4. Example: linear actions on vector spaces. Suppose that X = Cn,endowed with the standard Kahler structure, and that K acts on X in a linearway and respecting the standard Hermitian form. Let ξ : k → EndCn be theinfinitesimal action. Then a moment map is given, for any v ∈ X, by

μ(v) = − i

2ξ∗(v ⊗ v∗).

So if s ∈ k then

〈μ(v), s〉 = − i

2〈v ⊗ v∗, ξ(s)〉 = − i

2〈v, ξ(s)v〉 ∈ R.

Hence, if X =⊕

Vj is the decomposition of X defined as Vj = Ker(ξ(s)− λj Id),where {λj} are the eigenvalues of ξ(s) then setting

X(s)− =⊕λj≤0

Vj

we have:

λx(es) =

{0 if x ∈ X(s)−,∞ if x /∈ X(s)−.

2.5. Example: projective spaces. Suppose that X = CPn = P(Cn+1) isendowed with the Fubini–Study symplectic form, and that K acts on X through alinear representation on Cn+1 by unitary automorphisms. Let ξ : k → EndCn+1 bethe infinitesimal action. A moment map μ : X → k∗ can be described as follows.Let x ∈ X be any point and let v ∈ Cn+1 \ {0} be any vector representing x; then

μ(x) = − i

2ξ∗

(v ⊗ v∗

‖v‖2

).

So for any s ∈ k we have 〈μ(v), s〉 = − i2 〈v, ξ(s)v〉/‖v‖2 ∈ R. Let V =

⊕Vη be the

decomposition in eigenspaces of ξ(s), so that ξ(s) acts on Vη as multiplication byη. Since ξ(s) is skew Hermitian, Vη can only be nonzero if η is purely imaginary.Let v =

∑vη, with vη ∈ Vη, and define τ ∈ iR by the property that iτ = sup{iη |

vη �= 0}. Assume that |s| = 1. A simple computation gives:

λx(es) = −iτ/2 = iτ/2.

Now assume that there exists a real number σ > 0 such that all eigenvalues of σξ(s)belong to iZ. Then one can define a one parameter subgroup

α : C∗ → GL(n+ 1,C)

by the condition that α(eit) = exp(tσξ(s)), and the set of characters of α is equalto {−iση | Vη �= 0}. It follows that x0 := limz→0 α(z) · x is the point in CPn

represented by vτ . The character of C∗ acting on the fiber of OPn(−1) over x0

is equal to −iστ . The latter integer, which we denote by μO(1)(x;α), is what isusually called the maximal weight in GIT (see for example the paragraph beforeTheorem 3.9 in [T]). It follows that

(2.10) λx(es) = −σμO(1)(x;α)/2.

Every one parameter subgroup α can be obtained applying the previous construc-tion to a suitable s. Conversely, it is easy to check that for any s ∈ k satisfying|s| = 1 there is some one parameter subgroup α such that (2.10) holds (althoughin general its relation with s might not be as simple as in the case considered ear-lier). The general case of projective actions, as described in Section 1.2, can be

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reduced to the previous one by taking a linearization of the action. Then the previ-ous computations suggest that there should be a purely algebraic Hilbert–Mumfordcriterion for polystability (what is missing in the previous discussion is the algebraicanalogue of the condition on two points in the boundary at infinity of K\G beinggeodesically connected) . This statement, due to Schmitt, is given and proved inthe appendix to this paper.

2.6. Example: actions on Grassmannians. We now generalize the pre-vious computation. Assume that X is equal to Gr(k, n), the Grassmannian ofk-dimensional subspaces of Cn. As Kahler structure we take on X the restriction

of the Fubini–Study Kahler structure via the Plucker embedding X ↪→ CP (nk )−1.

Consider the action of U(n) on X induced by the fundamental representation ofU(n) on Cn. We identify the elements of X with orthogonal projections π ∈ EndCn

whose image is k-dimensional. Then the moment map is given, for any s ∈ u(n),by

μ(π)(s) = − i

2Tr(π ◦ s).

Consider the decomposition Cn =

⊕rj=1 Vj in eigenspaces of s, where s acts on Vj

as multiplication by λj . A simple (but instructive) computation gives the followingformula for the maximal weight: if V ∈ Gr(k, n), then

λV (es) =1

2

⎛⎝λr dimV +

r−1∑j=1

(λj − λj+1) dim(V ∩ Vj)

⎞⎠ .

Note that Grassmannians are particular cases of flag manifolds, so the formula formaximal weights given by Theorem 1.4 also includes them. We hence have twodifferent ways of describing maximal weights of Grassmannians: the previous one,which is explicit and concrete, and the one given by Theorem 1.4, which is moresynthetic and geometric but also more abstract.

3. Flag manifolds: maximal weights and Tits distance

In this section we prove Theorem 1.4. To begin with, let us recall the definitionof the Tits distance and of the relevant Kahler structures on flag manifolds.

As always in this paper, K denotes a compact Lie group and G its complexifi-cation. Let k be the Lie algebra of K. Denote by B : k× k → R a negative definitesymmetric pairing, invariant under the adjoint action of K on k. If K is semisimplethen B can be for example the Killing pairing: B(u, v) = Tr(adu ad v).

3.1. Tits distance. Define, for any u, v ∈ k, the positive definite pairing

〈u, v〉 = −B(u, v).

This gives rise to a G-invariant Riemannian metric on K\G, which allows one totalk about angles between tangent vectors.

Given any point x ∈ K\G and any y ∈ ∂∞(K\G) we define the tangent vector

v(x, y) ∈ Tx(K\G)

by the property that the geodesic ray γ : [0,∞) → K\G satisfying γ(0) = x andγ′(0) = v(x, y) represents y.

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The Tits distance between two points y, y′ ∈ ∂∞(K\G) is defined as

dT (y, y′) = sup

x∈K\GAngle(v(x, y), v(x, y′)),

where the angle takes values in [0, π]. For example, if there exists a geodesic con-verging on one side to y and on the other side to y′, then dT (y, y

′) = π. Theconverse is also true, but this is not at all obvious, since the Tits distance is definedby taking the supremum over a noncompact space. A proof can be given combiningthe computation in Lemma 3.1 and the results in Section 5.2 of [M].

3.2. Symplectic geometry of flag manifolds. There are different possibleways of looking at flag manifolds of G. From the point of view of symplecticgeometry it is natural to identify them with coadjoint orbits in k∗. To be precise, ifX = G/P is a flag manifold then to identify X with a coadjoint orbit is exactly thesame thing as choosing a K-invariant symplectic form on X and a moment mapμ : X → k∗: indeed, μ(X) ⊂ k∗ is a coadjoint orbit because K acts transitively onX. The symplectic structure on the coadjoint orbits O∗ = μ(X) coincides with theKostant–Souriau symplectic form ΩO∗ , which is defined as follows: if σ ∈ O∗ ⊂ k∗

and u, v ∈ TσO∗ are tangent vectors then we can write u = ad∗(a)(σ), v = ad∗(b)(σ)

for some elements a, b ∈ k, and we have

ΩO∗(u, v) = σ([a, b]).

In our discussion, however, we prefer to identify flag manifolds with adjointorbits in k instead of coadjoint orbits in k∗. Using the invariant pairing B we get aK-

equivariant isomorphism of vector spaces δ : k−→ k∗, which puts in correspondence

adjoint orbits with coadjoint orbits. If O ⊂ k is an adjoint orbit, the Kostant–Souriau symplectic form translates into a symplectic form ωO = δ∗ΩO∗ on O (hereO∗ = δ(O)), which can be described as follows. Let y ∈ O and let α, β ∈ TyO. Wecan write α = [a, y] and β = [b, y] for some a, b ∈ k, and we have

ωO(α, β) = B([a, b], y).

A moment map μ : O → k∗ of the action of K on O can be computed as follows: ify ∈ O and a ∈ k then

(3.11) 〈μ(y), a〉 = B(y, a).

To understand the converse construction identifying adjoint orbits O ⊂ k withflag manifolds, it is useful to use as a bridge the G-orbits in ∂∞(K\G), which can beidentified both with adjoint orbits and with flag manifolds. For these identificationsto be G-equivariant, however, we need to invert the action of G on ∂∞(K\G), whichwas previously defined as an action on the right. So for any y ∈ ∂∞(K\G) and anyg ∈ G we define

g · y := y · g−1.

If O is an adjoint orbit, then picking any u ∈ O and setting s = u/|u| we have thefollowing two (well known) facts:

(1) the G-orbit and theK-orbit through es in ∂∞(K\G) are the same: G·es =K · es (see for example Lemma 5.2 in [M]);

(2) the stabilizer of es in G is a parabolic subgroup P ⊂ G (see the beginningof Section 5.1 in [M]).

123


It follows that the orbit G · es ⊂ ∂∞(K\G) is G-equivariantly isomorphic to G/P .Denote by φ : G · es → G/P a G-equivariant isomorphism. Then the map f : O →G/P defined as f(k ·u) = φ−1(k · es) is a K-invariant isomorphism, so the action ofG on G/P can be transported to an action on O which extends the adjoint action.Furthermore, G/P has a natural complex structure which combines with ωO to givea Kahler structure.

The previous discussion implies that if we take any x ∈ X and we denote byu ∈ k the point corresponding to x via the identification of X with an adjoint orbit,then the maximal weight function is given by

(3.12) λ(X,ωO)x (ev) = lim

t→∞B(u · e−itv, v)

for any v ∈ S(k).

3.3. Comparing maximal weights and Tits distance. Taking (3.12) intoaccount, Theorem 1.4 is equivalent to the following lemma:

Lemma 3.1. Assume that y, y′ ∈ ∂∞(K\G), and that y = eu, y′ = ev withu, v ∈ S(k). Then

− cos dT (y, y′) = lim

t→∞B(u · e−itv, v).

Proof. An immediate consequence of the definition of the action of G on∂∞(K\G) is the following formula: for any x ∈ K\G, y ∈ ∂∞(K\G) and g ∈ G,

(3.13) v(x · g, y) = Dρg(v(x, y · g−1)),

where ρg is as in the proof of Lemma 2.1. Using (3.13) we have

− cos dT (y, y′) = − inf

g∈G〈u · g, v · g〉

= supg∈G

−〈u · g, v · g〉

= supg∈G

B(u · g, v · g).

Since v · e−itv = v for any t, we have

supg∈G

B(u · g, v · g) ≥ limt→∞

B(u · e−itv, v · e−itv) = limt→∞

B(u · e−itv, v).

We now prove the converse inequality. Let X denote the orbit G · y ⊂ ∂∞(K\G),endowed with the Kahler structure ωO described in Section 3.2 (so O is the adjointorbit through u), and denote by μ : X → k∗ the moment map of the action of Kon X. Let g ∈ G be any element. We have:

limt→∞

B(u · e−itv, v) = λ(X,ωO)y (ev) by formula (3.12)

= λ(X,ωO)g−1·y (ev · g) by Lemma 2.1

≥ 〈μ(g−1 · y), v · g〉 by formula (2.7)

= B(u · g, v · g) by formula (3.11).

We deduce that

supg∈G

B(u · g, v · g) = limt→∞

B(u · e−itv, v),

so the lemma is proved. �

124


Corollary 1.5 follows immediately from Lemma 3.1. Indeed, we may assume,as in the statement of Lemma 3.1, that y = eu, y

′ = ev, and (using the action ofsome element in G if necessary) we can also assume that ξ(t) = [eitu]. But then

− cosAngle(ξ′(t), η′t(0)) = B(u · e−itv, v),

so by Lemma 3.1 we have

− cos dT (y, y′) = lim

t→∞− cosAngle(ξ′(t), η′t(0)).

Since both the Tits distance and the angle on the right hand side take values in[0, π], we can apply arccos to both sides of the equality and obtain Corollary 1.5.

3.4. Relation to the work of Kapovich, Leeb and Millson. We havealready mentioned that some of the notions and results of this paper also appear,applied to the case of flag manifolds, in the paper [KLM] by Kapovich, Leeb andMillson, although often the same concepts receive a different name and the proofsfollow a different scheme. We now briefly explain these analogies and differences.

(1) The integral of the moment map in the case of flag manifolds is a Buse-man function; this follows from the formula for the derivative of Busemanfunctions (see formula (4) in [KLM]).

(2) It follows that the asymptotic slope, defined in (7) of [KLM], coincideswith the maximal weight function for flag manifolds.

(3) In view of the preceding, it is clear that formula (10) in [KLM] is com-pletely parallel to Theorem 1.4. But the logic of the proof of (10) in[KLM] is different from ours, since Kapovich et al. rely on the propertyof Tits distance stated in Corollary 1.5, whereas our arguments are basedon the general equivariance property of the maximal weight function (andwe do not use any property of Tits distance apart from its definition).

(4) As we mentioned in Section 1.2, in the case of projective manifolds themaximal weight function is essentially the same thing as the maximalweights in GIT. So Theorem 1.4 establishes a relation between Tits dis-tance and GIT on flag manifolds. In Section 4 of [KLM] such a relationis proved for the case X = CPn.

(5) Finally, the notion of nice semistability introduced in Definition 3.13 of[KLM] is very likely equivalent to the notion of analytic polystabilityintroduced in [M].

Appendix: An algebraic proof for Mundet i Riera’s polystabilitycriterion in GIT

A.H.W. Schmitt

Dedicated to Professor S. Ramanan on the occasion of his 70th birthday

A.1. Statement of the result. Let G be a reductive linear algebraic groupover the algebraically closed field k and ρ : G −→ GL(V ) a representation on thefinite dimensional k-vector space V . A point v ∈ V is called semistable, if theclosure of its G-orbit in V does not contain the origin, and polystable, if v �= 0 andits orbit G ·v is closed in V . Note that a polystable point is semistable. We remindthe reader of the following.

125


Theorem A.1 (Hilbert–Mumford criterion). Let v ∈ V and v ∈ G · v be a pointwith closed G-orbit. Then, there exists a one parameter subgroup λ : Gm(k) −→ Gwith

limz→∞

λ(z) · v ∈ G · v.

Given a one parameter subgroup λ : Gm(k) −→ GL(V ) and γ ∈ Z, we definethe eigenspace to the weight γ as

V (γ) :={v ∈ V |λ(z) · v = zγ · v ∀z ∈ Gm(k)

}.

Let γ1 < · · · < γs+1 be the weights with non-trivial eigenspaces. The vector spaceV then decomposes as

V = V (γ1)⊕ · · · ⊕ V (γs+1).

For a point v ∈ V \ {0}, we define

μ(v, λ) := max{γi | v has a non-trivial component in V (γi)

}.

We can restate the Hilbert–Mumford criterion as follows.

Corollary A.2. i) A point v ∈ V is semistable, if and only if

μ(v, λ) ≥ 0

holds for every one parameter subgroup λ : Gm(k) −→ G.ii) A point v ∈ V is polystable, if and only if it is semistable and for every

one parameter subgroup λ : Gm(k) −→ G with μ(v, λ) = 0 there is a group elementg ∈ G with

limz→∞

λ(z) · v = g · v.

Remark A.3. Let v ∈ V be a semistable point and λ : Gm(k) −→ G a oneparameter subgroup with μ(v, λ) = 0. Then,

v∞ := limz→∞

λ(z) · v

is the component of v in the weight space V (0) and a semistable point.

For a parabolic subgroup Q ⊂ G, there is a Levi subgroup L ⊂ Q, i.e., areductive subgroup, such that

Q = Ru(Q)� L,

where Ru(Q) is the unipotent radical of Q. Any two Levi subgroups L and L′ ofQ are conjugate by an element r ∈ Ru(Q).

Example A.4. i) For a one parameter subgroup λ : Gm(k) −→ G,

QG(λ) :={g ∈ G | lim

z→∞λ(z) · g · λ(z)−1 exists in G

}is a parabolic subgroup of G and every parabolic subgroup of G is obtained in thisway.

The unipotent radical of QG(λ) is

Ru

(QG(λ)

)=

{g ∈ G | lim

z→∞λ(z) · g · λ(z)−1 = e

},

and

LG(λ) :={g ∈ G |λ(z) · g = g · λ(z) ∀z ∈ Gm(k)

}is a Levi subgroup of QG(λ).

126


For a one parameter subgroup λ : Gm(k) −→ GL(V ) with weights γ1 < · · · <γs+1, we define the (partial) flag

V•(λ) : {0} � V1 := V (γ1) � V2 :=

:= V (γ1)⊕ V (γ2) � · · · � Vs := V (γ1)⊕ · · · ⊕ V (γs) � V.

Then, QGL(V )(λ) is the GL(V )-stabilizer of the flag V•(λ), LGL(V )(λ) identifieswith GL(V (γ1))×· · ·×GL(V (γs+1)), and Ru(QGL(V )(λ)) consists of those matriceswhose diagonal blocks are unit matrices.

The latter description implies the following: If v ∈ V is a point with μ(v, λ) = γiand g ∈ Ru(QGL(V )(λ)), then v and g · v have the same component in V (γi).

ii) By the conjugacy of Levi subgroups, we can obtain every pair (Q,L), consist-ing of a parabolic subgroup Q of G and a Levi subgroup L of Q, as (QG(λ), LG(λ))for an appropriate one parameter subgroup λ : Gm(k) −→ G.

iii) For every point v ∈ V , every one parameter subgroup

λ : Gm(k) −→ G,

and every g ∈ QG(λ), one has, by i),

μ(v, λ) = μ(v, g · λ · g−1).

Two parabolic subgroups Q and Q′ are called opposite, if Q ∩ Q′ is a Levisubgroup of both Q and Q′. Likewise, two one parameter subgroups λ and λ′ aresaid to be opposite1, if QG(λ) and QG(λ

′) are opposite parabolic subgroups of G.In [M], Mundet i Riera proves a characterization of polystable points in a

symplectic manifold with respect to a Hamiltonian action by a compact real group.Specialized to the GIT setting, his result reads as follows.

Theorem A.5 (Mundet i Riera). A point v ∈ V is polystable, if and onlyif it is semistable and for every one parameter subgroup λ : Gm(k) −→ G withμ(v, λ) = 0 there is an opposite one parameter subgroup λ′ with

μ(v, λ′) = 0.

A.2. A Strengthening of the Hilbert–Mumford Criterion.

Proposition A.6. A point v ∈ V is polystable, if and only if it is semistableand for every one parameter subgroup λ : Gm(k) −→ G with μ(v, λ) = 0 there is agroup element g ∈ Ru(QG(λ)) with limz→∞ λ(z) · v = g · v.

The proof rests on the following.

Proposition A.7 (Kraft/Kuttler). Let G be a reductive affine algebraic groupand H ⊂ G a reductive closed subgroup. Let v ∈ G/H a point and λ : Gm(k) −→ Ga one parameter subgroup, such that v∞ := limz→∞ λ(z) · v exists. Then, thereexists an element g ∈ Ru(QG(λ)) with v∞ = g · v.

Proof. We refer to [GLSS], Proposition 2.1.2, p. 1188, or [S], Proposition2.4.2.5, p. 181. For a different proof and a generalization to perfect fields, thereader may consult Theorem 3.4 in [BMRT]. �

1Mundet i Riera does, as Mumford, require that λ−1 is conjugate to λ′. The proof we presenthere works also for that definition.

127


Proof of Proposition A.6. The stated condition is clearly sufficient. Tosee that it is also necessary, let v ∈ V be a polystable point. Its stabilizer Gv isreductive. If Char(k) = 0, then the orbit map ov : G −→ V , g �−→ g · v induces anisomorphism G/Gv −→ G · v. So, the result follows directly from Proposition A.7with H = Gv.

In positive characteristic, the orbit map ov induces a bijective and propermorphism G/Gv −→ G · v, so that the same reasoning applies. �

A.3. Proof of Theorem A.5. Let v ∈ V be a polystable point and

λ : Gm(k) −→ G

a one parameter subgroup with μ(v, λ) = 0. Then, v∞ := limz→∞ λ(z) ·v exists, byRemark A.3. Proposition A.6 grants the existence of an element g ∈ Ru(QG(λ))

with v∞ = g · v. Note that λ stabilizes v∞. Thus, λ := g−1 · λ · g stabilizes v. We

set λ′ := λ−1. Obviously,

μ(v, λ′) = 0, QG(λ) = QG(λ), and LG(λ′) = QG(λ

′) ∩QG(λ) = LG(λ).

Let us assume, to the converse, that the point v ∈ V satisfies Mundet i Riera’scriterion. We will verify that v satisfies the property stated in Proposition A.6,too. So, let λ : Gm(k) −→ G be a one parameter subgroup with μ(v, λ) = 0 andset v∞ := limz→∞ λ(z) · v. By assumption, there is a one parameter subgroupλ′ : Gm(k) −→ G with μ(v, λ′) = 0 which is opposite to λ. Taking into accountExample A.4, iii), and the conjugacy of Levi subgroups, we may assume(

QG(λ′), LG(λ

′))=

(QG(λ

′), QG(λ′) ∩QG(λ)

).

Likewise, we find an element g ∈ Ru(QG(λ)), such that(QG(g · λ · g−1), LG(g · λ · g−1)

)=

(QG(λ), QG(λ) ∩QG(λ

′)).

Now, λ′ and g · λ · g−1 lie in the connected component of the identity of the centerof L := LG(λ

′) = LG(g · λ · g−1). In particular, there is a maximal torus T of Gwhich contains both λ′ and g · λ · g−1. It follows that

(A.14) sign(〈 g · λ · g−1, χ 〉

)= −sign

(〈λ′, χ 〉

), ∀χ ∈ X(T ).

On the other hand,

μ(v, λ′) = 0 = μ(v, λ)Example A.4, iii)

= μ(v, g · λ · g−1).

The latter equality is compatible with (A.14), if and only if v is contained in theeigenspace to the weight zero of both λ′ and g ·λ ·g−1. In particular, g ·λ ·g−1 fixesv or, equivalently, λ fixes g−1 ·v. We will be done, if we can check that v∞ = g−1 ·v.

Recall from Remark A.3 that v∞ is the component of v in the eigenspace ofλ for the weight zero. Noting that ρ maps QG(λ) into QGL(V )(λ), Example A.4,

i), shows that v, v∞, and g−1 · v have the same component in the weight zeroeigenspace of λ. Since both v∞ and g−1 · v are elements of that eigenspace, theydo agree. �

Remark A.8. Theorem A.5 implies Proposition A.7. Indeed, if λ′ is a oneparameter subgroup with μ(v, λ′) = 0 which is opposite to λ, then we find elements

g ∈ Ru(QG(λ)) and g′ ∈ Ru(QG(λ′)), such that LG(g

′ · λ′ · g′−1) = LG(g · λ · g−1).

Moreover,

μ(v, g′ · λ′ · g′−1) = μ(v, λ′) = 0 = μ(v, λ) = μ(v, g · λ · g−1).

128


Again, this is only possible, if v lies in the weight space to weight zero of g ·λ · g−1.Then, g−1 · v is fixed by λ and as before we infer

g−1 · v = limz→∞

λ(z) · v.

References

[B] W. Ballmann, Lectures on spaces of nonpositive curvature, With an appendix by MishaBrin DMV Seminar 25, Birkhauser Verlag, Basel, 1995.

[BMRT] M. Bate, B. Martin, G. Roehrle, R. Tange, Closed orbits and uniform S-instability inInvariant Theory, preprint arXiv:0904.4853.

[BJ] A. Borel, L. Ji, Compactifications of Symmetric and Locally Symmetric Spaces, Mathe-matics: Theory & Applications, Birkhauser, Boston (2006).

[E] P. Eberlein, Structure of manifolds of nonpositive curvature, Global differential geometryand global analysis 1984 (Berlin, 1984), 86–153, Lecture Notes in Math. 1156 Springer,Berlin, 1985.

[GLSS] T.L. Gomez, A. Langer, A.H.W. Schmitt, I. Sols, Moduli spaces for principal bundles inarbitrary characteristic, Adv. Math. 219 (2008), 1177–1245.

[HS] P. Heinzner, G.W. Schwartz, Cartan decomposition of the moment map, Math. Ann.337 (2007), 197–232.

[KLM] M. Kapovich, B. Leeb, J. Millson, Convex functions on symmetric spaces, side lengthsof polygons and stability inequalities for weighted configurations, J. Differential Geom.81 (2009), no. 2, 297–354.

[MFK] D. Mumford, J. Fogarty, F. Kirwan, Geometric Invariant Theory, 3rd edition, Erg.Math., Springer Verlag (1994).

[M] I. Mundet i Riera, A Hilbert–Mumford criterion for polystability in Kahler geometry,Trans. Amer. Math. Soc., to appear.

[S] A.H.W. Schmitt, Geometric Invariant Theory and Decorated Principal Bundles, ZurichLectures in Advanced Mathematics, European Mathematical Society (EMS), vii+389pp.

[T] R.P. Thomas, Notes on GIT and symplectic reduction for bundles and varieties, Surveysin differential geometry X, 221–273, Surv. Differ. Geom., 10, International Press (2006).

Departament d’Algebra i Geometria, Facultat de Matematiques, Universitat de

Barcelona, Gran Via de les Corts Catalanes 585, 08007 Barcelona, Spain


Freie Universitat Berlin, Arnimallee 3, 14195 Berlin, Germany


129


Orthogonal Bundles Over Curves in Characteristic Two

Christian Pauly

Dedicated to S. Ramanan on his 70th birthday

Abstract. Let X be a smooth projective curve of genus g ≥ 2 defined overa field of characteristic two. We give examples of stable orthogonal bundleswith unstable underlying vector bundles and use them to give counterexamplesto Behrend’s conjecture on the canonical reduction of principal G-bundles forG = SO(n) with n ≥ 7.

Let X be a smooth projective curve of genus g ≥ 2 and let G be a connectedreductive linear algebraic group defined over a field k of arbitrary characteristic.One associates to any principal G-bundle EG over X a reduction EP of EG to aparabolic subgroup P ⊂ G, the so-called canonical reduction — see e.g. [Ra], [Be],[BH] or [H] for its definition. We only mention here that in the case G = GL(n) thecanonical reduction coincides with the Harder-Narasimhan filtration of the rank-nvector bundle associated to EG.

In [Be] (Conjecture 7.6) K. Behrend conjectured that for any principal G-bundle EG over X the canonical reduction EP has no infinitesimal deformations,or equivalently, that the vector space H0(X,EP ×P g/p) is zero. Here p and g arethe Lie algebras of P and G respectively. Behrend’s conjecture implies that thecanonical reduction EP is defined over the same base field as EG.

We note that this conjecture holds for the structure groups GL(n) and Sp(2n)in any characteristic, and also for SO(n) in any characteristic different from two —see [H] section 2. On the other hand, a counterexample to Behrend’s conjecturefor the exceptional group G2 in characteristic two has been constructed recently byJ. Heinloth in [H] section 5.

In this note we focus on SO(n)-bundles in characteristic two. As a startingpoint we consider the rank-2 vector bundle F∗L given by the direct image under theFrobenius map F of a line bundle L over the curve X and observe (Proposition 4.4)

2010 Mathematics Subject Classification. 14H60, 14H25.The author was partially supported by the Ministerio de Educacion y Ciencia (Spain) through

the grant SAB2006-0022.


1



131

2 CHRISTIAN PAULY

that the SO(3)-bundle A := End0(F∗L) is stable, but that its underlying vectorbundle is unstable and, in particular, destabilized by the rank-2 vector bundleF∗OX . We use this observation to show that the SO(7)-bundle

F∗OX ⊕ (F∗OX)∗ ⊕A

equipped with the natural quadratic form gives a counterexample to Behrend’sconjecture. Replacing A by A = End(F∗L) we obtain in the same way a coun-terexample for SO(8), and more generally for SO(n) with n ≥ 7 after adding directsummands of hyperbolic planes. Note that Behrend’s conjecture holds for SO(n)with n ≤ 6 because of the exceptional isomorphisms with other classical groups.

The first three sections are quite elementary and recall well-known facts onquadratic forms, orthogonal groups and their Lie algebras, as well as orthogonalbundles in characteristic two. In the last section we give an example (Proposition6.1) of an unstable SO(7)-bundle having its canonical reduction only defined afteran inseparable extension of the base field.

This note can be considered as an appendix to the recent paper [H] by J.Heinloth, whom I would like to thank for helpful discussions. I also thank thereferee for useful suggestions and, in particular, for an improvement of the proof ofProposition 4.4.

1. Quadratic forms in characteristic two

The purpose of this section is to recall definitions and basic properties of qua-dratic forms in characteristic two.

Let k be a field of characteristic two and let V be a vector space of dimensionn over k. We denote by Sym2V and Λ2V the symmetric and exterior square of V ,i.e.,

Sym2V = V ⊗ V/I, and Λ2V = V ⊗ V/J ,

where I and J are the subspaces of V ⊗ V linearly generated by the tensors v ⊗w + w ⊗ v and v ⊗ v respectively, with v, w ∈ V . Note the inclusion I ⊂ J . Wedenote by F (V ) := V ⊗k k the Frobenius-twist of V . We observe that there existsa well-defined k-linear injective map

F (V ) ↪→ Sym2V, v ⊗ λ → λ(v · v),

where v ·v denotes the symmetric square of v. More precisely, we have the followingexact sequence of k-vector spaces

0 −→ F (V ) −→ Sym2V −→ Λ2V −→ 0.

Let σ denote the involution on V ⊗V defined by σ(v⊗w) = w⊗ v and let S2(V ) ⊂V ⊗ V denote the σ-invariant subspace. Then one has the exact sequence

0 −→ S2(V ) −→ V ⊗ V −→ Λ2V −→ 0,

where the last arrow denotes the canonical projection of V ⊗ V onto Λ2V . Bychoosing a basis of V , it can be checked that the image of S2(V ) under the canonicalprojection V ⊗ V → Sym2V equals the subspace F (V ).

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ORTHOGONAL BUNDLES OVER CURVES IN CHARACTERISTIC TWO 3

By definition, see e.g. [Bo] section 23.5, a quadratic form on the vector spaceV is a k-valued function on V such that

Q(av + bw) = a2Q(v) + b2Q(w) + abβ(v, w) a, b ∈ k; v, w ∈ V,

where β is a bilinear form on V . The bilinear form β associated to a quadratic formQ is determined by the formula

β(v, w) = Q(v + w) +Q(v) +Q(w).

Is is clear that β is symmetric, hence β can be regarded as an element in S2(V∗).

We denote the vector space of quadratic forms on V by Quad(V ) and introduce thek-linear map

Φ : Sym2(V ∗) −→ Quad(V ), ϕ · ψ → Q, with Q(v) = ϕ(v)ψ(v).

Here ϕ · ψ ∈ Sym2(V ∗) denotes the image of ϕ⊗ ψ ∈ V ∗ ⊗ V ∗ under the canonicalprojection V ∗ ⊗ V ∗ → Sym2(V ∗). By choosing a basis of V and using formula (5)of [Bo] section 23.5, it is easily seen that Φ is an isomorphism. We will thereforeidentify Quad(V ) with the symmetric square Sym2(V ∗).

The assignment Q → β gives rise to the k-linear polarisation map

P : Sym2(V ∗) −→ S2(V∗), Q → β.

Over a field k of characteristic different from 2 we recall that the polarisation mapP is an isomorphism. In our situation one can easily work out that

ker P = F (V ∗) and coker P = F (V ∗).

Quadratic forms Q ∈ ker P correspond to squares of linear forms on V and thelast equality asserts that the bilinear form β is alternating, i.e., β(v, v) = 0 for allv ∈ V .

We say that Q is non-degenerate if β induces an isomorphism β : V → V ∗ for neven, and if dim ker β = 1 and Q| ker β = 0 for n odd. We say that a linear subspace

W ⊂ V is isotropic for Q if Q|W = 0.

2. Orthogonal groups and their Lie algebras in characteristic two

In this section we work out in detail the structure of the Lie algebras of theorthogonal groups SO(7) and SO(8) over a field k of characteristic two, as well asof some of their parabolic subgroups P ⊂ SO(n). Following the principle of [H], wedescribe the representation so(n)/p of a Levi subgroup L ⊂ P , which we will uselater in section 5. The main reference is [Bo] section 23.6.

By definition the orthogonal group O(n) is the subgroup of GL(n) = GL(V )stabilizing a non-degenerate quadratic form Q ∈ Sym2(V ∗). Note that in char-acteristic two O(n) ⊂ SL(n). We also recall that, if n is odd, the group O(n) isconnected. If n is even, O(n) has two connected components distinguished by theDickson invariant ([D] page 301). In order to keep the same notation as in charac-teristic = 2, we denote by SO(n) the connected component of O(n) containing theidentity.

133

4 CHRISTIAN PAULY

2.1. The group SO(8). We consider V = k8 endowed with the non-degeneratequadratic form

Q = x1x7 + x2x8 + x3x5 + x4x6 ∈ Sym2(V ∗).

We choose this non-standard quadratic form in order to obtain a simple descriptionof the Lie algebra p of the parabolic subgroup P fixing an isotropic 2-plane, whichwill appear in section 5. The bilinear form β associated to Q is given by the matrix⎛

⎜⎜⎝O O O IO O I OO I O OI O O O

⎞⎟⎟⎠ ,

where I denotes the identity matrix(

1 00 1

)

and O the zero 2 × 2 matrix. The

computation carried out in [Bo] section 23.6 shows, after a renumbering of the linesand columns, that the Lie algebra so(8) ⊂ End(V ) of SO(8) consists of the matricesof the form

(2.1)

⎛⎜⎜⎝

X1 X2 X3 D1

X4 X5 D2tX3

X6 D3tX5

tX2

D4tX6

tX4tX1

⎞⎟⎟⎠ ,

where the Xi are 2× 2 matrices, the tXi denote their transpose, and the Di denote

matrices of the form(

0 λλ 0

)

.

We consider the maximal parabolic subgroup P ⊂ SO(8) preserving the iso-tropic 2-plane W ⊂ V = k8 defined by the equations xi = 0 for i = 3, . . . , 8. Itsorthogonal space W⊥ is then defined by the equations x7 = x8 = 0. The Levisubgroup L ⊂ P is isomorphic to GL(2)× SO(4). The Lie algebra p of P consistsof the matrices (2.1) that satisfy the conditions

X4 = X6 = 0 and D4 = 0.

The vector space so(8)/p sits naturally in the exact sequence of L-modules

(2.2) 0 −→ Hom(W,W⊥/W ) −→ so(8)/p −→ D −→ 0,

where D is the one-dimensional subspace generated by the element e ⊗ f + f ⊗e ∈ W ∗ ⊗ W ∗ = Hom(W,W ∗) ∼= Hom(W,V/W⊥), where e and f are linearlyindependent.

2.2. The group SO(7). We consider V = k7 endowed with the non-degeneratequadratic form

Q = x1x6 + x2x7 + x3x4 + x25 ∈ Sym2(V ∗).

The bilinear form β associated to Q is given by the 7× 7 matrix⎛⎜⎜⎝

O O 0 IO I 0 O0 0 0 0I O 0 O

⎞⎟⎟⎠ .

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The radical ker β of Q is generated by e5. A straightforward computation showsthat the Lie algebra so(7) ⊂ End(V ) of SO(7) consists of the matrices of the form

(2.3)

⎛⎜⎜⎝

X1 X2 0 D1

X3 Δ 0 tX2

A1 A2 0 A3

D2tX3 0 tX1

⎞⎟⎟⎠ ,

where the Ai are 2× 1 matrices and Δ is a diagonal matrix of the form(

λ 00 λ

)

. We consider the maximal parabolic subgroup P ⊂ SO(7) preserving the isotropic2-plane W ⊂ V = k7 defined by the equations xi = 0 for i = 3, . . . , 7. Its orthogonalspace W⊥ is then defined by the equations x6 = x7 = 0. The Levi subgroup L ⊂ Pis isomorphic to GL(2) × SO(3). The Lie algebra p of P consists of the matrices(2.3) that satisfy the conditions

X3 = 0, A1 = 0, and D2 = 0.

As above we have an exact sequence of L-modules

(2.4) 0 −→ Hom(W,W⊥/W ) −→ so(7)/p −→ D −→ 0.

3. Orthogonal bundles

By definition an orthogonal bundle over a smooth projective curve X definedover a field k is a principal O(n)-bundle over X, which we denote by EO(n). Equiv-alently a O(n)-bundle corresponds to a pair (E , Q), where E is a rank-n vectorbundle and Q is a non-degenerate OX -valued quadratic form on E , i.e., an el-ement Q ∈ H0(X, Sym2(E∗)), such that at every point x ∈ X the quadraticform Qx ∈ Sym2(E∗

x) is non-degenerate. By abuse of notation we will writeEO(n) = (E , Q).

Since the parabolic subgroups of O(n) are the stabilizers of isotropic flags in kn,one observes that the notion of (semi-)stability of the O(n)-bundle EO(n) translatesinto the following condition on the pair (E , Q) (see e.g. [Ram] Definition 4.1): wesay that (E , Q) is semi-stable (resp. stable) as an orthogonal bundle if and only if

μ(F) :=degFrkF ≤ 0 (resp. < 0)

for all isotropic subbundles F ⊂ E .

A noteworthy result is the following

Proposition 3.1 ([Ram] Proposition 4.2). Assume that the characteristic ofk is different from two. The pair (E , Q) is semi-stable as an orthogonal bundle ifand only if the underlying vector bundle E is semi-stable.

In the next section we will show that the assumption on the characteristic cannot be removed.

4. Stable orthogonal bundles with unstable underlying bundle

We denote by F : X → X the absolute Frobenius of the curve X of genus g ≥ 2defined over an algebraically closed field k of characteristic two. We denote by KX

the canonical line bundle of X.

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6 CHRISTIAN PAULY

4.1. An orthogonal rank-3 bundle. Let E be a rank-2 vector bundle ofarbitrary degree over X. The determinant endows the bundle End0(E) of tracelessendomorphisms of E with an OX -valued non-degenerate quadratic form

det : End0(E) −→ OX .

Its associated bilinear form β is given by β(v, w) = Tr(v ◦w) for u, v local sectionsof End0(E) and the radical of det equals the line subbundle OX ↪→ End0(E) ofhomotheties.

Lemma 4.1. There is an exact sequence

(4.1) 0 −→ OX −→ End0(E) −→ F ∗(E)⊗ (detE)−1 −→ 0,

where the first homomorphism is the inclusion of homotheties into End0(E).

Proof. Consider V = k2 and W = End0(V ) equipped with the determinant.Note that for any g ∈ GL(V ) the conjugation Cg : W → W is orthogonal andleaves the identity Id invariant. Thus we obtain group homomorphisms GL(V ) →SO(W ) → GL(W/〈Id〉), g → Cg → Cg. A straightforward computation shows thatthe composite map is given by g → 1

det gF (g), where F (g) is the Frobenius-twist of

g. �Proposition 4.2. If E is stable, then (End0(E), det) is stable as an orthogonal

bundle.

Proof. It suffices to observe that an isotropic line subbundle M of End0(E)corresponds to a nonzero homomorphism φ : E → E⊗M−1 of rank 1. If L denotesthe line bundle imφ ⊂ E ⊗ M−1, we obtain the inequalities μ(E) < degL <μ(E ⊗M−1), which implies degM < 0. �

Let B denote the sheaf of locally exact differentials [Ray], which can be definedas the cokernel

(4.2) 0 −→ OX −→ F∗OX −→ B −→ 0,

of the inclusion OX ↪→ F∗OX given by the Frobenius map. Note that in char-acteristic 2 the sheaf B is a theta-characteristic, i.e., B2 = KX . We denote bye ∈ Ext1(B,OX) = H1(X,B−1) the non-zero extension class determined by theexact sequence (4.2). By tensoring this exact sequence with B−1 and taking thelong exact sequence of cohomology spaces, we obtain the following

Lemma 4.3. The extension class e generates the one-dimensional kernel of theFrobenius map, i.e.,

〈e〉 = ker(F : H1(X,B−1) −→ H1(X,K−1

X )).

In particular, we have a direct sum

F ∗ (F∗OX) = OX ⊕KX .

Let L be a line bundle of arbitrary degree. Then the rank-2 vector bundle F∗Lhas determinant equal to B ⊗ L, is stable and is destabilized by pull-back by theFrobenius map F ([LP]). More precisely, the bundle F ∗(F∗L)⊗ B−1 ⊗ L−1 is theunique non-split extension (see e.g. [LS])

(4.3) 0 −→ Bι−→ F ∗(F∗L)⊗B−1 ⊗ L−1 π−→ B−1 −→ 0.

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Proposition 4.4. (i) The vector bundle A := End0(F∗L) does not de-pend on L.

(ii) The vector bundle A is the unique non-split extension

0 −→ F∗OXφ−→ A ψ−→ B−1 −→ 0.

We denote by φ and ψ generators of the one-dimensional spacesHom(F∗OX ,A) and Hom(A, B−1).

(iii) The restriction of the quadratic form det to the subbundle F∗OX equalsthe evaluation morphism F ∗F∗OX −→ OX . In particular the restrictionof β to F∗OX is identically zero.

(iv) Let x and y be local sections of F∗OX and A. Then the two bilinearforms β(φ(x), y) and 〈x, ψ(y)〉 on the product F∗OX ×A differ by a non-zero multiplicative scalar. Here 〈., .〉 denotes the standard pairing betweenF∗OX and its dual (F∗OX)∗.

Proof. First we consider the exact sequence (4.1) for the bundle F∗L

0 −→ OX −→ End0(F∗L) −→ F ∗(F∗L)⊗B−1 ⊗ L−1 −→ 0.

We then observe that there exists a natural inclusion

F∗End(L) = F∗OX ↪→ End0(F∗L),

which extends the inclusion of homotheties OX ↪→ End0(F∗L). Note that thesubbundle F∗OX corresponds to endomorphisms of F∗L which are F∗OX -linear. Alocal computation shows that these are trace-free. Since detF∗OX = B, we obtainan exact sequence

0 −→ F∗OX −→ End0(F∗L) −→ B−1 −→ 0.

We denote by a its extension class in Ext1(B−1, F∗OX) = H1(X, (F∗OX) ⊗ B).We have (F∗OX) ⊗ B = (F∗OX)∗ ⊗ KX and by Serre duality H1(X, (F∗OX) ⊗B) = H0(X,F∗OX)∗ = H0(X,OX)∗, which implies that dimExt1(B−1, F∗OX) =1. Hence there exists (up to isomorphism) a unique non-split extension of B−1 byF∗OX . In order to show assertions (i) and (ii) it will be enough to check that a = 0.But the push-out of a under the map F∗OX → B gives the exact sequence (4.3),which is a non-split extension. This proves the claim. The proof of the equalitiesdimHom(F∗OX ,A) = dimHom(A, B−1) = 1 is standard.

The restriction of the bilinear form β to F∗OX is identically zero, since βfactorizes through the line bundle quotient B and β is alternating (see section 1).Hence the restriction of the quadratic form det lies in F ∗(F∗OX)∗ ⊂ Sym2(F∗OX)∗,i.e., corresponds to an OX -linear map α : F ∗F∗OX = OX ⊕KX → OX . We notethat α|OX

= id and α|KX= 0. This proves (iii).

Finally we observe that the bilinear form (x, y) → β(φ(x), y) factorizes throughthe quotient B × B−1, since β is identically zero on F∗OX by part (iii) and sincethe radical of β equals OX . On the other hand the bilinear form (x, y) → 〈x, ψ(y)〉also factorizes through B × B−1. This proves (iv), since both non-zero bilinearforms factorize through the unique (up to a multiplicative scalar) perfect pairingon B ×B−1. �

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8 CHRISTIAN PAULY

Since F∗OX is stable and μ(F∗OX) = g−12 > 0, we deduce that the Harder-

Narasimhan filtration of A is

0 ⊂ F∗OX ⊂ A.

However, since F∗L is stable, Proposition 4.2 implies that the orthogonal bundle(A, det) is stable.

Remark. Similarly, it can be shown that the rank-4 vector bundle A := End(F∗L)does not depend on the line bundle L, that it fits into the exact sequence

0 −→ F∗OX −→ End(F∗L) −→ (F∗OX)∗ −→ 0,

giving its Harder-Narasimhan filtration, and that (A, det) is a stable orthogonalbundle.

4.2. An orthogonal rank-4 bundle. Besides the example given in the lastremark of section 3.1, we can consider the unstable decomposable rank-4 vectorbundle

B = F∗OX ⊕ (F∗OX)∗

and endow it with the OX -valued quadratic form

Q(x+ x∗) = q(x) + q∗(x∗) + 〈x, x∗〉,where x, x∗ are local sections of the rank-2 bundles F∗OX and (F∗OX)∗ respec-tively. The bracket 〈x, x∗〉 denotes the standard pairing between F∗OX and itsdual (F∗OX)∗ and q, q∗ are projections onto OX of the direct sums (see Lemma4.3)

(4.4) F ∗ (F∗OX) = OX ⊕KX , F ∗ (F∗OX)∗ = OX ⊕K−1X .

It is clear that Q is non-degenerate and that Q restricts to q and q∗ on the directsummands F∗OX and (F∗OX)∗ respectively.

Proposition 4.5. The orthogonal rank-4 bundle (B, Q) is stable.

Proof. Let M be an isotropic line subbundle of F∗OX ⊕(F∗OX)∗ and assumethat degM ≥ 0. By stability of (F∗OX)∗ the line bundle M is contained in F∗OX ,which gives by adjunction a nonzero map F ∗M → F ∗F∗OX → OX , contradictingisotropy of M . Next, let S be an isotropic rank-2 subbundle of F∗OX ⊕ (F∗OX)∗

and assume μ(S) ≥ 0. By the previous considerations S is stable, hence there is nonon-zero map S → (F∗OX)∗. Therefore S is a rank-2 subsheaf of F∗OX , but thiscontradicts isotropy of S. �

Remark. Clearly the two bundles A and B are non-isomorphic.

5. Counterexamples to Behrend’s conjecture

We consider the rank-7 vector bundle

E = F∗OX ⊕A⊕ (F∗OX)∗

equipped with the non-degenerate quadratic form Q defined by

Q(x+ y + x∗) = 〈x, x∗〉+ det y,

where x, x∗ and y are local sections of F∗OX , (F∗OX)∗ and A respectively. Notethat the SO(7)-bundle ESO(7) = (E , Q) has a reduction to the Levi subgroup

L = GL(2)× SO(3) ⊂ P ⊂ SO(7)

138


and that its L-bundle EL given by the pair (F∗OX ,A) is stable, since F∗OX andA are stable GL(2)− and SO(3)-bundles respectively. Since μ(F∗OX) > μ(A) = 0,we obtain that the canonical reduction of the unstable SO(7)-bundle (E , Q) is givenby the P -bundle EP := EL ×L P . Using the exact sequence of L-modules (2.4) weobtain the equality

Hom(F∗OX ,A) = H0(X,EP ×P so(7)/p).

Note that H0(X,EP ×P D) ⊂ Hom(F∗OX , (F∗OX)∗) = 0 by stability of F∗OX .But the former space is nonzero by Proposition 4.4 (ii). Hence this provides acounterexample to Behrend’s conjecture.

The following unstable SO(8)-bundles also provide counterexamples to Beh-rend’s conjecture

F∗OX ⊕ A ⊕ (F∗OX)∗ and F∗OX ⊕ B ⊕ (F∗OX)∗,

with the quadratic form given by the standard hyperbolic form on the two sum-mands F∗OX⊕(F∗OX)∗ and the quadratic forms on A and B introduced in sections4.1 and 4.2. Similarly we use the exact sequence (2.2) and Proposition 4.4 (ii) toshow that the corresponding spaces H0(X,EP ×P so(8)/p) are non-zero.

Remark. One can work out the relationship between the unstable SO(7)-bundle(E , Q) and the unstable G2-bundle EG2

constructed in [H] section 5: first, we recallthat G2 ⊂ GL(6). Consider the symplectic rank-6 bundle E obtained from (E , Q)under the purely inseparable group homomorphism SO(7) → Sp(6) ⊂ GL(6). Thenthe bundle E has a reduction to EG2

.

6. Non-rationality of the canonical reduction of an SO(7)-bundle

This section is largely inspired by the last remark of [H]. We consider theunstable SO(7)-bundle (E , Q) introduced in section 5. We denote by K the fieldk(t), by XK the curve X ×k K and by EK the vector bundle over XK obtained aspull-back of E under the field extension K/k. We will consider the quadratic form

QK on the bundle EK defined by

QK(x+ y + x∗) = 〈x, x∗〉+ det y + tq(x),

where x, x∗ and y are local sections of F∗OX , (F∗OX)∗ and A respectively and qis the quadratic form on F∗OX defined in (4.4). Note that by Proposition 4.4 (iii)one has q(x) = det(φ(x)).

Proposition 6.1. (i) The quadratic form QK is non-degenerate.

(ii) The canonical reduction of the SO(7)-bundle (EK , QK) over XK is notdefined over K, but only over the inseparable quadratic extension K ′ =K[s]/(s2 − t).

Proof. We consider the vector bundle EK′ over XK′ obtained from EK underthe field extension K ′/K and introduce the automorphism gs of EK′ defined by thematrix ⎛

⎝ IdF∗OX0 0

sφ IdA 00 sψ Id(F∗OX)∗

⎞⎠ .

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10 CHRISTIAN PAULY

We choose the generators φ ∈ Hom(F∗OX ,A) and ψ ∈ Hom(A, B−1) such thatβ(φ(x), y) = 〈x, ψ(y)〉 for any local sections x, y of F∗OX and A respectively — seeProposition (4.4) (iv). Note that gs is an involution. Then we have

gs(x, y, x∗) = (x, y + sφ(x), x∗ + sψ(y)).

We also denote by Q and QK the quadratic forms on EK′ obtained from Q and QK

under the field extensions K ′/k and K ′/K respectively. Then we have the equality

(6.1) Q(gs(x, y, x∗)) = Q(x, y, x∗) + s2q(x) = QK(x, y, x∗),

i.e., the quadratic forms Q and QK on EK′ differ by the automorphism gs. In

particular QK is non-degenerate and the pair (EK , QK) determines an SO(7)-bundleover the curve XK .

Because of (6.1) the canonical reduction of the SO(7)-bundle (EK , QK) is givenby the isotropic rank-2 vector bundle α : F∗OX ↪→ EK′ with α(x) = (x, sφ(x), 0)for any local section x of F∗OX . It is clear that the inclusion α is not defined overK. �

References

[Be] K. A. Behrend, Semi-stability of reductive group schemes over curves, Math. Ann. 301(1995), 281–301.

[BH] I. Biswas, Y. Holla, Harder-Narasimhan reduction of a principal bundle, Nagoya Math. J.174 (2004), 201–223.

[Bo] A. Borel, Linear algebraic groups, Graduate Texts in Mathematics 106, Second Edition,Springer-Verlag (1991)

[D] J. Dieudonne, Algebraic homogeneous spaces over fields of characteristic two, Proc. Amer.Math. Soc. 2 (1951), 295–304.

[H] J. Heinloth, Bounds for Behrend’s conjecture on the canonical reduction, Int. Math. Res.Notices (2008), Vol. 2008, article IDrnn 045.

[LP] H. Lange, C. Pauly, On Frobenius-destabilized rank-2 vector bundles over curves, Comm.Math. Helv. 83 (2008), 179–209.

[LS] H. Lange, U. Stuhler, Vektorbundel auf Kurven und Darstellungen der algebraischen Fun-damentalgruppe, Math. Zeit. 156 (1977), 73–83.

[Ram] S. Ramanan, Orthogonal and spin bundles over hyperelliptic curves, Proc. Indian Acad.Sci. 90 (1981), no. 2, 151–166.

[Ra] A. Ramanathan, Moduli for principal bundles, Algebraic Geometry Proceedings, Copen-hagen 1978, Lect. Notes Math. 732, Springer.

[Ray] M. Raynaud, Sections des fibres vectoriels sur une courbe, Bull. Soc. Math. France 110(1982), no.1, 103–125.

Departement de Mathematiques, Universite de Montpellier II - Case Courrier 051,

Place Eugene Bataillon, 34095 Montpellier Cedex 5, France


140


The Atiyah-Singer Index Theorem

Raghunathan

Abstract. In this paper we give an account of the Atiyah-Singer Index The-orem following the ideas in the original announcement in the Bulletin of theAmerican Mathematical Society. There are however, significant differences inthe way several steps in the proof are handled.

1. Introduction

This is essentially an expository account of the Atiyah-Singer Index theorem,undoubtedly one of the great theorems of the twentieth century. The theoremestablishes the equality of two numbers that are associated with an elliptic lineardifferential operator on a smooth compact manifold, one defined with the geometricdata on the operator and the other by using analysis. For the precise statementof the theorem, we refer to the next section. We give here a proof of the theorembroadly following the lines of the original proof given by Atiyah and Singer in theirannouncement in [2] offering however some new ways for dealing with the varioussteps that lead up to the final proof. Atiyah and Singer did not publish the detailsof the proof indicated in their paper. The details were however worked out in theseminar conducted by Palais at the Institute for Advanced Study, the notes of whichare published in the Annals of Mathematics Studies series [4].

The proof given here follows in broad outline the ideas in the Bulletin an-nouncement; however it deviates in many ways in the matter of details indicatedthere (the Palais seminar follows closely the scheme set out in [2] even in the matterof details). Odd-dimensional manifolds however are dealt with exactly as in [4] -both the analytic and topological indices are zero for all differential operators onodd-dimensional manifolds.

The first important difference is in the proof of the all-important fact that theanalytic index of a linear elliptic differential operator depends only on the K-theoryelement defined by its symbol. The proof given here makes no use of the theoryof pseudo-differential operators - there is a trade-off though, in that we need toappeal to some somewhat more refined results from topology - among others, tothe fact that the odd dimensional sphere is rational homotopy equivalent to thereal projective space of the same dimension under the natural map.

2010 Mathematics Subject Classification. 58J20.Key words and phrases. Elliptic operators, Index.


1



141

M. S.

2 M.S.RAGHUNATHAN

The second point of difference is that we make use of some qualitative infor-mation on the heat kernel to prove the “bordism invariance” of the analytic index(which is a crucial ingredient of the proof) rather than results about boundary valueproblems in the theory of partial differential equations as is done in [4]. Specificallywe need the fact that the terms in the asymptotic expansion of the heat kernel ofa second order elliptic operator are determined by the local behaviour of the op-erator. This is essentally a result of Minakshisundaram and Pleijel [3]. It may beremarked that the heat equation plays a crucial role in many of the different proofsof the index theorem; however the use of the heat kernel in those proofs has a verydifferent flavour from our use of it here.

Atiyah and Singer define an equivalence relation involving cobordism on theset consisting of elliptic operators (on all smooth manifolds and vector bundles onthem); they give the set of equivalence classes a natural ring structure. Then theyshow that both the analytic and topological indices are constant on equivalenceclasses of elliptic operators (this is where the “bordism invariance” mentioned inthe last paragraph is needed) and the resulting Q-valued functions are both homo-morphisms of this ring into Q. They analyse the structure of the ring and exhibita set of generators; the index theorem for these generators is a consequence onthe one hand of the Gauss-Bonnet theorem and on the other hand Hirzebruch’ssignature theorem. We replace these arguments by an induction argument on thedimension of the manifold. This is achieved by viewing the indices as functions on(even) cohomology with coefficients in Q rather than on K-theory tensored with Q

and make use of a theorem due to Serre [5] (as well as an idea employed by Thom ina different context). The theorem in question asserts that, if X is a finite complexof dimension n and α is a q-cohomology class with coefficients in Q with n < 2q−1,then there is a map f from X to Sq such that α is in the image of Hq(Sq,Q) underthe map induced by f .

Finally it may be remarked that the index theorem itself is formulated in [2]for elliptic pseudo-differential operators (for which too the topological and analyticindices are defined), but once the theorem for differential operators is established thegeneral case follows from the following considerations: in the even-dimensional case,the K-theoretic symbol of an elliptic pseudo-differential operator (which determinesboth the indices) is theK-theoretic symbol of a suitable elliptic differential operator;the odd-dimensional case can be reduced to the even-dimensional case by formingthe product with the unit circle on which there is a pseudo-differential operator forwhich analytic and topological indices are both 1 while both these indices behavemultiplicatively with respect to the formation of the (external) tensor products ofthe symbols and the Kunneth isomorphism in K-theory.

This account confines itself to differential operators. As the remarks in thelast paragraph indicate this yields the theorem in the more general case of pseudo-differential operators as well.

My introduction to the index theorem took place in a seminar organised byRamanan and M.S.Narasimhan soon after the Bulletin announcement of Atiyahand Singer appeared, in which I participated. When I was thinking about thatseminar (during a spell of nostalgia a few years ago) this somewhat different wayof handling the proof occured to me. This paper is the result and I am happy thatit appears in this volume in honour of Ramanan.

M. S. RAGHUNATHAN142

THE ATIYAH-SINGER INDEX THEOREM 3

I thank Peter Newstead for reading the manuscript carefully and correctingerrors.

2. The statement of the theorem

2.1. The symbol and the analytic index. Throughout this paper (exceptin the Appendix) we will be working with smooth manifolds and bundles. Unlessotherwise specified explicitly all maps considered will be smooth. In particularsections of vector bundles will be smooth. Let M be a smooth compact closedmanifold of dimension m. Let E and F be complex vector bundles on M . Recallthat a linear differential operator from E to F is a C-linear map D : Γ(E) → Γ(F )such that, for any section Φ of E, the support of D(Φ) is contained in the support ofΦ. All differential operators considered in this paper will be linear; so we will dropthe adjective “linear” in the sequel. If we denote by Jk(E) the bundle of k-jets of E,then, for all suitably large k, D defines and is defined by a bundle homomorphismfrom Jk(E) to F . The minimal k for which this homomorphism is defined is theorder of D. Let T (resp. T ∗) be the tangent (resp. cotangent) bundle of M andSk(T ∗) the k-th symmetric power of T ∗; then one has an inclusion of Sk(T ∗)⊗ Ein Jk(E). If now D is a differential operator of order k then the homomorphism itdefines from Jk(E) to F gives by restriction a homomorphism from Sk(T ∗)⊗E to F .The diagonal inclusion of T ∗ in Sk(T ∗) enables one to view this last homomorphismas a bundle homomorphism of the pull-back p∗(E) of E to the pull-back p∗(F ) ofF under the natural projection p : T ∗ → M . This element of Hom(p∗(E), p∗(F ))which on each fibre of T ∗ is a homogeneous polynomial of degree k is the symbol,σ(D), of D. It is a basic fact from the theory of linear differential operators thatthe kernel (resp. cokernel) of an operator D whose symbol is an injective (resp.surjective) homomorphism outside the zero section of T ∗ is finite-dimensional. Adifferential operator D (from E to F ) is elliptic if and only if the symbol of D isan isomorphism outside the zero section of T ∗. This means of course that E andF have the same rank. The analytic index a(D) of D is defined as the integerdim(kernel(D))− dim(cokernel(D)).

2.2. The K-theoretic symbol. We fix a Riemannian metric on M and de-note by B (resp. S) the unit disc (resp. sphere) subbundle of T ∗. We denote byp the projection of T ∗ on M as well as its restriction to B. If now D is an ellipticoperator from E to F its symbol σ(D) defines an isomorphism of the restrictionsto S of the bundles p∗(E) and p∗(F ) (here for a bundle V on M , p∗(V ) is thepull-back of V to B under p). The “difference construction” in K-theory (see [4],p.15) now yields an element σ0(D) in the relative K-group K0(B,S) which we willrefer to as the K-theoretic symbol of D in the sequel. The following properties ofthe K-theoretic symbol follow from its definition (via the difference construction).

Lemma 2.1. If D : Γ(E) → Γ(F ) and D′ : Γ(F ) → Γ(G) are elliptic differentialoperators from E to F and from F to G respectively, then D′D is an elliptic operatorfrom E to G and and σ0(D

′D) = σ0(D) + σ0(D′). Also if the operator D (resp.

D′) is an elliptic operator from E (resp. E′) to F (resp. F ′) and D ⊕ D′ is theoperator from E ⊕ E′ to F ⊕ F ′defined by setting

(D ⊕D′)(σ ⊕ σ′) = D(σ)⊕D(σ′)

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4 M.S.RAGHUNATHAN

for sections σ, σ′ respectively of E and E′, then D ⊕D′ is elliptic and

σ0(D ⊕D′) = σ0(D) + σ0(D′).

2.3. The topological index. Now, we have a natural ring homomorphism,the Chern character (see [4], p.14) Ch, from K0(B,S) to the sum Heven(B,S;Q) ofthe even-dimensional singular cohomology groups of the pair (B,S) with coefficientsin Q (which in fact gives an isomorphism from K0(B,S)⊗Q to Heven(B,S;Q)).Weget thus a cohomology class Ch(σ0(D)) in the latter group. Let Td(M) denote theTodd class of M as well as its pull-back to H∗(B;Q). The cup-product givesa bilinear pairing H∗(B,S;Q) × H∗(B;Q) → H∗(B,S;Q) and we thus obtaina cohomology class Ch(σ0(D)).Td(M) in H∗(B,S;Q). Now the pair (B,S) hasa canonical orientation and hence H2n(B,S;Z), which is isomorphic to Z, hasa canonical generator. The topological index t(D) of the elliptic operator D isdefined as the (rational) number obtained by evaluating Ch(σ0(D).Td(M) on thisgenerator. With these definitions and notation we can state the Atiyah-Singer indextheorem.

Theorem 2.2. Let D be an elliptic operator on a compact manifold from avector bundle E to a vector bundle F . Then a(D) = t(D).

2.4. Remarks. Note that the theorem implies that the analytic index of Ddepends only on the K-theoretic symbol of D (as this is true for t(D) by its verydefinition). This fact is proved as the first step in the proof of the theorem. Alsonote that the topological index, which we know only to be a rational number, turnsout to be an integer as a consequence of the theorem.

3. Construction of some differential operators

3.1. For a vector bundle W on M and a point x in M , Wx will denote the fibreof W at X. Let E and F be vector bundles on M and D a differential operatorfrom E to F . We introduce a Riemannian metric on M and denote by μ the Borelmeasure defined by it on M . We also intoduce hermitian inner products along thefibres of E and F . In the sequel these inner products on E and F as well as theinner products on T and T ∗ defined by the Riemannian metric will be denoted<,>. With this notation, we have the notion of the adjoint D∗ of the operator:this is the unique differential operator from F to E which satisfies the followingcondition: for sections α of E and β of F ,∫

M

< D(α), β > dμ =

∫M

< α,D∗(β > dμ.

That such a D∗ exists and has the same order as D is a standard fact and is provedeasily by integration by parts on local charts over which the bundles E and F aretrivial. Also the adjoint D∗∗ of D∗ is D. When D is elliptic, so is D∗ and one hasnatural isomorphisms of kernel D (resp. cokernel D) on cokernel D∗ (resp. KernelD∗). It follows that a(D) = −a(D∗). In particular, if F is the same as E and Dis self-adjoint, i.e. D = D∗, then a(D) = 0. The symbol of the operator Δ = D∗Dfrom E to itself assigns to each v in T ∗

x , x ∈ M , the automorphism σ∗(v)σ(v) ofEx, where for an endomorphism U of Ex, U

∗ denotes its conjugate transpose withrespect to the inner product on Ex. It follows that σ(D∗D) is at every point of Sa hermitian-symmetric positive definite automorphism and we can therefore raiseit to the power t for every t in the closed interval [0,1]. This yields a homotopy



between the symbol of D∗D and the identity automorphism of p∗(E) over S. Asthis last automorphism (evidently) extends over all of B, we conclude (from thedefinition of the difference construction) that the K-theoretic symbol of D∗D iszero. We have the following proposition.

Proposition 3.1. Given any vector bundle E on a manifold M there is a self-adjoint elliptic operator ΔE from E to E of order 2 such that σ0(ΔE) is zero (anda(ΔE) = 0).

3.2. In the light of the disscussion in the last paragraph, to prove the proposi-tion we need only exhibit an operator D of order 1 from E to a suitable bundle Fsuch that the symbol σ(D) is an injective bundle homomorphism outside the zerosection of T ∗ – one can then take for Δ the operator D∗D for some hermitian innerproducts along the fibres of E and F and a Riemannian metric on M . To constructa D of order 1, we take F to be Hom(T,E) (= T ∗ ⊗ E), the bundle of E-valued1-forms on M . We fix a hermitian inner product on E and a Riemannian metricon E; these give rise to a natural inner product on F as well. We then take D tobe the exterior differentiation with respect to a unitary connection ω on E. Onesees then that for this operator Δ, the symbol associates to each v in T ∗

x , x ∈ M ,the automorphism || v ||2.Identity.

3.3. For a vector bundle V and a non-negative integer r, we denote by r.V thedirect sum of r copies of V . A differential operator D from E to another vectorbundle F evidently defines a differential operator r.D from r.E to r.F which oneach component of r.E is the operator D from E to F , the latter considered as thecorresponding component of F . It follows from the definition of the K-theoreticsymbol that σ0(r.D) = r.σ0(D). With this notation we will now establish thefollowing.

Proposition 3.2. For a positive integer n, let λ(n) = 22n+2. Then, given anyvector bundle E on a manifold M of dimension m, there is an elliptic operator DE

of order 1 from λ(m).E to itself such that a(DE) = 0 and σ0(DE) = 0.

3.4. Let I ′ : M → R2m+1 be an imbedding and I the imbedding in R2m+2

obtained by composing the standard inclusion of R2m+1 in R2m+2 with I ′. Wehave then an inclusion of T in the trivial real vector bundle on M of rank 2m+ 2and hence (using the standard inner product on R

2m+2) an inclusion of T ∗ in thetrivial real vector bundle of rank 2m + 2. Now let V denote the trivial complexvector bundle of rank 2m + 2. As I factors through the inclusion of R2m+1 inR2m+2, we note that T ∗ is contained in a trivial (complex) vector sub-bundle V ′

such that V is the direct sum of V ′ and a trivial line bundle equipped with aneverywhere non-zero section s. Now let x ∈ M and v ∈ T ∗

x . Define a homomorphisme(v) : T ∗

x ⊗ Λp(V )x → Λp+1(V )x (resp. i(v) : T ∗x ⊗ Λp(V )x → Λp−1(V )x) (where

for a non-negative integer l, Λl(V ) is the lth exterior power of V ) as the exterior(resp. interior) multiplication by v (note that T ∗

x is a subspace of Vx and V beingthe trivial bundle carries a natural inner product). Let Λe(V ) (resp. Λo(V )) bethe direct sum of the even (resp. odd) exterior powers of V . Then e(v) + i(v) as vvaries in T ∗ defines homomorphisms T ∗ ⊗ V e → V o and T ∗ ⊗ V o → V e which wedenote σe and σo respectively in the sequel. If now E is any vector bundle on M ,σe ⊗ (Identity) is a homomorphism from p∗(V e ⊗ E) to p∗(V o ⊗ E) which is anisomorphism outside the zero-section: this is because σo⊗(Identity)·σe⊗(Identity)

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6 M.S.RAGHUNATHAN

is the endomorphism || v ||2 .(Identity) of p∗(V e ⊗ E) (here v ∈ T ∗x , x ∈ M). Now

if we introduce a hermitian inner product on E and a connection on E compatiblewith it, one sees that σe ⊗ (Identity) and σo ⊗ (Identity) can be lifted to (elliptic)differential operators DE and D∗

E which are adjoints of each other. It follows that

the direct sum D of DE and D∗E gives an elliptic self-adjoint operator of order 1

from Λ(V )⊗ E (Λ(V ) is the exterior algebra bundle of V ) to itself. Moreover

t(e(v) + i(v)) + (1− t)(e(s(x)) + i(s(x))) (v ∈ T ∗x , x ∈ M)

gives a homotopy over S of the symbol of D with an isomorphism which extendsover all of T ∗. This proves the proposition: since V is a trivial bundle of rank2m + 2, Λ(V ) is trivial of rank λ(m) and Λ(V ) ⊗ E is isomorphic to a direct sumof λ(m) copies of E.

4. K-theoretic symbol and the analytic index

Our aim in this section is to establish the following:

Theorem 4.1. Let D and D′ be elliptic operators from vector bundles E andE′ to F and F ′ respectively. If the K-theoretic symbols of D and D′ are equal, thena(D) = a(D′).

4.1. We will first prove the following much weaker statement:

Proposition 4.2. Suppose given two elliptic differential operators D and D′,both from E to F of the same order k, such that there is a smooth homotopy σt,t ∈ [0, 1], of isomorphisms from p∗(E) to p∗(F ) such that each σt is a homogeneouspolynomial of degree k along the fibres of T ∗ and σ0 = σ(D), σ1 = σ(D′). Thena(D) = a(D′). In particular if D and D′ have the same symbol, a(D) = a(D′).

4.2. For any vector bundle W on M with a hermitian inner product and anon-negative integer k, one defines a pre-Hilbert structure on Γ(W ) with the innerproduct<,>k defined as follows. Fix once and for all a finite covering {Ui | i ∈ I}of M by cooordinate open sets and an isomorphism of the restriction of W to theUi with a trivial bundle. Fix a shrinking {Vi | i ∈ I} of {Ui | i ∈ I}. Then eachsection of W defines a collection F = {Fi | i ∈ I}, each Fi being a Rn-valuedsmooth function on Vi. For a pair F, F′ of sections of W , let

< F,F′ >k=∑

i∈I, |α|≤k

< ∂αFi/∂xα, ∂α/∂xα > .

The completion of this pre-Hilbert structure on Γ(W ) is denoted Hk(W ). Thetopological vector space structure on Hk(W ) is independent of the choice of thecovering, its shrinking and the trivialisations of W over the open sets of the cover-ing.

We now fix vector bundles E and F onM . With the notation introduced above,one then sees that a differential operator D : Γ(E) → Γ(F ) of order k extends toa continuous linear map from Hk(E) to Hk(F ). When D is elliptic, there is aconstant C = C(D) > 0 such that, for all σ in the orthogonal complement ofkernel(D), one has || D(σ) || ≥ C(D). || σ ||. Suppose now that Dt, t ∈ [0, 1], isa smooth family of elliptic operators. Let t0 ∈ [0, 1] and let H ′ be the orthogonalcomplement of kernel(Dt0) in Hk(E). Then there is a neighbourhood U of t0 in[0,1] and a constant C > 0 such that C(Dt) ≥ C for all t in U . From this it follows



easily that a(Dt) is independent of t. Now if D and D′ have the same symbol,t.D + (1 − t)D′, t ∈ [0, 1], provides a smooth family of elliptic operators leadingto the conclusion that a(D) = a(D′). Now since the inclusion of the pull-back ofSk(T ∗) ⊗ E to M × [0, 1] in the pull-back of Jk(E) admits a right inverse vectorbundle morphism, we see that, given σt as in the statement of the proposition, wecan find a smooth family of (elliptic) operators Dt, t ∈ [0, 1], such that σ(Dt) = σt.It follows now that a(D) = a(D′).

4.3. Suppose now thatD is an elliptic differential operator from a vector bundleE to a vector bundle F . It is obvious that t(r.D) = r.t(D) and a(r.D) = r.a(D)for any non-negative integer r. Thus equality of a(D) and t(D) holds if it holds fora(r.D) and t(r.D) for some integer r greater than zero. Observe next that, if D′ isan elliptic operator from F to a vector bundle G of the same rank, then a(D′D) =a(D′) + a(D) and t(D′D) = t(D′) + T (D). Now suppose that D (resp. D′)is anelliptic operator from E to F (resp. from E′ to F ′) of order k (resp. k′) are suchthat their K-theoretic symbols are the same. We want to show that they have thesame analytic index. Observe first that we may replace the operators D and D′ byr.D and r.D′ for any positive integer r. By choosing r = λ(n) as in Proposition4 and composing with a suitable power of DF (resp. D′

F ), we may assume that k= k′ = 2l is an even integer. Let W (resp. W ′) be a vector bundle such that thedirect sum of E (resp. E′) and W (resp. W ′) is trivial of rank q. Let ΔE and Δ′

E

be as in Proposition 3 and let L and L′ denote their respective lth powers. Let D

(resp. D′) be the direct sum of D (resp. D′) and L (resp. L′). Clearly D and D′

have the same K-theoretic symbol and we need to prove that their analytic indicesare the same.

4.4. This means that we can assume that we are in the following situation. LetE denote the trivial bundle of rank q and D (resp. D′) be an elliptic operator fromE to a rank-q vector bundle F (resp. F ′) of even order 2l such that σ0(D) = σ0(D

′).We have to show that a(D) = a(D′). Since E is trivial, so is F (resp F ′) over S.This means that the bundles F and F ′ are stably isomorphic: this follows fromthe fact that the K-theoretic symbols of D and D′ obtained by the diifferenceconstruction are the same. Using Proposition 4 for the trivial bundle, one findsthat by forming the direct sum with a suitable number of copies of the trivial linebundle 1 equipped with a suitable power of Δ1, we may assume that both D andD′ are elliptic operators from the (trivial) bundle E to the same bundle F .

4.5. That the two operators have the same K-theoretic symbol implies thefollowing: there exists a positive integer N and an automorphism u : E⊕1N → E⊕1N such that Σ1 = (σ(D)⊕IdN ).u and Σ2 = σ(D′)⊕IdN are homotopic through a1-paramaeter family Σt, t ∈ [0, 1] of sections ofHom(p∗(E)⊕1N , p∗(F )⊕1N ) over S,each Σt being an isomorphism. Σ1 and Σ2 are both symbols of differential operatorsD1 and D′

1 respectively of order k = 2l (see Proposition 3 and the discussion in 3.2)with the sameK-theoretic symbols asD andD′ respectively; further a(D1) = a(D′

1)as u is a 0th order operator and therefore has index 0. Now let P denote the realprojective space bundle associated to T ∗: this is a quotient of S by the involutiveautomorphism which maps each unit vector in S into its negative. Let q denote thenatural projection of P onM . Then since the order k of D and D′ is even, σ(D) andσ(D′) are pull-backs of sections σ and σ′ of Hom(q∗(E), q∗(F )). It is shown in theAppendix that the existence of the homotopy σt implies that, replacing D1 and D′

1

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8 M.S.RAGHUNATHAN

by r.D1 and r.D′1, we can assume that Σ1 and Σ2 are homotopic through sections

of Hom(q∗(E), q∗(F )) which are all isomorphisms of q∗(E) on q∗(F ). Equivalently,we can assume that the homotopy Σt above is such that, for v ∈ T ∗

x , x ∈ M , Σt(v)= Σt(−v).

4.6. Fix some extension of the homotopy Σt to a homotopy of sections over

all of T ∗ of Hom(p∗(E), p∗(F )) and denote this extension by Σt. We assume that

the Σt vanish outside a compact neighbourhood of S in T ∗. Now convolving Σt

with the function (c/π)1/2exp(−c || (.) ||2) on T ∗ along the fibres of T ∗ we obtain a

homotopy Σ(c,t) such that, on every fibre of T ∗, it admits a Taylor expansion which

converges uniformly on compact sets. As c tends to ∞, Σ(c,t) converges uniformly

on compact sets to Σt. It follows that, for all large c, Σ(c,t) is invertible over S asalso is a suitable truncation θ(c,t) of its Taylor expansion (along the fibres). Letτ(c,t) = (θ(c,t) + θ(c,−t))/2. Since Σt is an even function on the fibres of T ∗, we canchoose c and the truncation so that τ(c,t) is invertible over all of S and τ(c,0) (resp.

τ(c,1)) equals Σ(c,0) (resp.Σ(c,1)). We replace Σt now by such a τ(c,t) and denote thelatter by Σt in the sequel.

4.7. It is clear that Σt is now a polynomial along the fibres of T ∗ with everyhomogeneous component of even degree. It can therefore be made homogeneouswith the function || . ||2 on T ∗ (which is homogeneous of degree 2 along the fibres ofT ∗ and is identically 1 on S). In other words we can assume that the homotopy σt

is a homogeneous polynomial function along the fibres of T ∗ of fixed degree. Thismay mean however that the degrees of Σ0 and Σ1 have been raised. However fromProposition 4 we know that the composites of ΔF⊕1N raised to any power with D1

and D′1 have the same analytic index as well as the same K-theoretic symbol as D

and D′ respectively. We are now in a situation where the homotopy between thesymbols of D and D′ is through isomorphisms of p∗(E) on p∗(F ) all of which arehomogeneous polynomials of the same even degree. Theorem 5 now follows fromProposition 6.

5. The signature operators

In this section we will describe the so-called twisted signature operators.

5.1. Let M be a manifold of dimension m. We fix a Riemannian metric onM which gives rise to a Borel measure μ on M . Let ΩM denote the line bundleof exterior m-forms on M . When there is no ambiguity about the underlyingmanifold M , we denote ΩM simply Ω. The metric gives a canonical reduction ofthe structure group R

∗ of the line bundle Ω to the subgroup {1,−1} so that a2-sheeted covering of M imbeds in Ω, the two sheeted covering being trivial or non-trivial according as M is orientable or not; in the former case an orientation givesa natural trivialisation of the covering. This reduction of structure group leadsalso to a canonical isomorphism of Ω ⊗ Ω with the trivial bundle M × R. If themanifold is oriented, the metric gives a natural trivialisation of Ω. For an integerp with 0 ≤ p ≤ m, let Ωp denote the p-th exterior power of T ∗; in particular Ωm

= Ω. The exterior multiplication then defines a pairing Ωp ⊗ Ωm−p → Ω whichis non-degenerate and hence gives a non-degenerate pairing of Ωp with Ωm−p ⊗ Ω.Using the isomorphism of Ωm−p with its dual given by the Riemannian metric, weobtain a bundle isomorphism from Ωp to Ωm−p⊗Ω which is denoted ∗ in the sequel.



The natural isomorphism from Ω⊗Ω to the trivial bundle yields (by taking tensorproduct with the identity morphism of Ω) an isomorphism from Ωp ⊗ Ω to Ωm−p

which will also be denoted ∗. One then finds that ∗2 on Ωp equals (−1)p(m−p).Suppose now that α and β are two p-forms on M ; then α∧∗β is a section of Ω⊗Ωand – as this last line bundle is naturally isomorphic to the trivial bundle – is afunction on M . On the other hand the Riemannian metric gives an inner productdenoted <,> on the Ωp. One checks easily that < α, β >= α ∧ ∗β. For α, β asabove we set

(α, β) =

∫M

< α, β > dμ

5.2. Suppose now that E is a complex vector bundle with a hermitian innerproduct along its fibres. Let ω be a unitary connection on E. We then have thedifferential operator of order 1, namely exterior differentiation dω with respect toω of E-valued p-forms on M – these are nothing but sections of the bundle Ωp⊗E:

dω : Γ(Ωp ⊗ E) → Γ(Ωp+1 ⊗ E).

The hermitian inner product on E gives a conjugate-linear isomorphism from E tothe dual E∗. Forming the tensor product of this isomorphism with ∗, we obtain aconjugate-linear isomorphism from Ωp⊗E to Ωm−p⊗Ω⊗E∗ which we denote also∗:

∗ : Ωp ⊗ E → Ωm−p ⊗ Ω⊗ E∗.

Now, if α (resp. β) is a p-form (resp. (m− p)-form) on M with values in E (resp.E∗), then using the pairing between E and E∗, we get a m-form α∧β with values inΩ⊗Ω. One checks easily that for p-forms α, β with values in E, < α, β > (the innerproduct on Ωp ⊗ E is the one deduced from that given by the Riemannian metricon M and the hermitian inner product on E) equals α ∧ ∗β (treated as a functionvia the canonical trivialisation of Ω⊗Ω). It follows moreover from Stokes’ theoremthat, for a p-form α and a p+ 1-form β, we have, setting δω = (−1)p+1 ∗ dω∗,∫

M

< dωα, β > dμ =

∫M

< α, δωβ > dμ.

Thus we see that δω is the adjoint of dω. Let Ωe (resp. Ωo) be the direct sum ofthe even (resp. odd) exterior powers of T ∗. Then dω + δω is an elliptic differentialoperator (of order 1) from Ωe ⊗ E to Ωo ⊗ Ω ⊗ E. In the sequel we denote thisoperator DE,deRh and refer to it as the de Rham-Hodge operator for E (or theE-twisted de Rham-Hodge operator). When E is the trivial line bundle, we dropthe E in the suffix and refer to it simply as the de Rham-Hodge operator.

5.3. When the dimension m of M is even, equal to 2n, say, there is anotherway of decomposing the bundle ΛT ∗ of (all) exterior differential forms into a directsum of two vector bundles Ω+ and Ω−: if we define ′∗ : Ωp → Ωm−p ⊗Ω by setting′ ∗ (ξ) = ip(p−1) ∗ (ξ), one checks easily that (′∗)2 = (−1)n (here i is a square rootof -1 fixed once and for all). It follows that when n is even (resp. odd), that ΛT ∗

decomposes as a direct sum of two subbundles Ω+ and Ω− where Ω+ (resp. Ω−)isthe eigen-subbundle of ΛT ∗ corresponding to the eigenvalue θ (resp. −θ) for ′∗, θbeing 1 or i according as n is even or odd. The operator dω + δω is then seen todefine an elliptic operator from Ω+ ⊗E to Ω− ⊗E (which will be denoted DE,ω inthe sequel); we call this the E-twisted signature operator on M . We have then thefollowing crucial fact from K-theory (for a proof see [4], pp. 225–26).

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10 M.S.RAGHUNATHAN

Theorem 5.1. Let Th : K0(M) → K0(B,S) be the homomorphism definedby setting Th(E) = σ0(DE,ω) for a vector bundle E on M . Then Th gives anisomorphism of K0(M)⊗Q on K0(B,S)⊗Q.

Note that for a vector bundle E, Th(E) depends only on the K-theoretic classof E and is independent of the choice of the hermitian metric and the connectionon E and defines a homomorphism from K0(M) to K0(B,S). Theorems 5 and7 together show that, to prove Theorem 2, it suffices to prove that a(DE,ω) =t(DE,ω). This is what will be done in the rest of this paper.

6. “Bordism Invariance” of the index

We adopt the following notation in the sequel. For a vector bundle E on amanifold M , a(E) (resp.t(E)) will be the analytic (resp. topological) index of theE-twisted signature operator DE,ω . This notation is justified as the indices inquestion are independent of the connection ω as the K-theoretic symbol of DE,ω

itself is independent of ω. We then have:

Theorem 6.1. Suppose now that M and M ′ are two smooth manifolds and Eand E′ are vector bundles on them. Suppose further that there is a manifold withboundary W such that the boundary of W is the disjoint union of M and M ′ andthere is a vector bundle E on W such that E restricts to E (resp. E′) on M (resp.M ′). Then a(E) = a(E′) and t(E) = t(E′).

6.1. For the result for the topological index which is easy to prove, see [4],p.228. For the analytic index we will make use of some results about the heatkernel essentially due to Minakshisundaram and Pleijel [3]. We recall these now.The operator DE,ω has for its adjoint the operator DE⊗Ω,′ω with ′ω denoting theconnection on E ⊗ Ω obtained from ω and the flat connection on Ω. We set ′E =E⊗ω in the sequel.Then D′E,′ωDE,ω (resp. DE,ωD′E,′ω) is an elliptic second orderoperator of Ω+⊗E (resp.Ω−⊗′E) to itself which we denote simply Δ+ (resp. Δ−) inthe rest of this section. For t ≥ 0, the operator exp(−tΔ+) (resp. exp(−tΔ−)) hasa smooth kernel K+(t, x, y) (resp.K−(t, x, y)), a smooth section of Hom(p∗1(Ω

+ ⊗E), p∗2(Ω

+⊗E)) (resp. Hom(p∗1(Ω−⊗′E), p∗2(Ω

−⊗′E)) which depends smoothly onthe parameter t as well. Further K+(t, x, x) (resp. K−(t, x, x)) has an asymptoticexpnasion

∑0≤r≤∞ tr−m/2K+

r−m/2(x, x) (resp.∑

0≤r≤∞ tr−m/2K−r−m/2(x, x)) as t

goes to zero. The analytic index of D+E,ω is then equal to∫

M

Trace(K+0 (x, x))dμ(x)−

∫M

Trace(K−0 (x, x))dμ(x)

The crucial fact we need about the asymptotic expansion is the following. If Eand E′ are two vector bundles with hermitian inner products <,> and <,>′ andunitary connections ω and ω′ respectively such that, on an open set U of M , wehave a hermitian isomorphism from the restriction of E to the restriction of E′

which carries ω into ω′, then, on U ×U , all the terms in the asymptotic expansionsof the two heat kernels coincide. In particular, the function Trace(K+

0 (x, x)) −Trace(K−

0 (x, x)) on M (which we will denote TE in the sequel) is determined inthe neighbourhood of any point by the local data on the Riemannian metric, thehermitian inner product on E and the unitary connection in that neighbourhood.TE is naturally a section of Ω ⊗ Ω and so is to be regarded as a closed m-form onM with values in Ω.



6.2. Since any cobordism between manifolds is a composition of successiveelementary cobordisms, we may assume for proving Theorem 8 that W is an ele-mentary cobordism. In other words, there is smooth function f : W → [−1, 1] suchthat M = f−1(−1), M ′ = f−1(1) and f has exactly one non-degenerate criticalpoint w (of index r, say) with f(w) = 0. One then has an open neighbourhood Uof w in W and a diffeomorphism F from an open disc D(4ε), ε > 0, of radius 4εaround the origin 0 in R

m, to U with F (0) = w and such that f composed with Fis the function Q defined on D(4ε) as follows: for a suitable decomposition of Rm

as an orthogonal direct sum of Rr and Rs (r + s = m), denoting by x1, x2, ....xr

(resp. y1, y2, .....ys) the coordinates in Rr (resp. Rs), one has

Q(x1, x2....xr, y1, y2, ....ys) =∑

1≤i≤s

y2i −∑

1≤j≤r

x2j .

For any c > 0, let D(c) denote the disc of radius c around the origin in Rm.

Introduce a Riemannian metric g′ on W such that it induces the standard metricon D(3ε) under the map F . Let X be the vector field on W ′ = W \ {w} suchthat < X, Y >= 0 for all vector fields Y on W ′ with Y f = 0 and < X,X > =1. The local 1-parameter group φt of local diffeomorphisms of W determined byX then defines for 0 < c < 3ε a smooth map Φ : (M \ F (D(c))) × [0, 1] → Wgiven by Φ(x, t) = φ2t(x) which is a diffeomorphism onto a closed subset W (c) inW ; the interior of the manifold with boundary (M \ F (D(3ε)))× [0, 1] maps ontothe interior of W (c) as a subset of W . This last open set is also diffeomorphicto (M ′ \ F (D(c))) × [0, 1] under the map that takes (x′, t) to φ−2t(x

′). We nowreplace the metric g′ by a metric g which coincides with g′ on F (D(ε)) and induceson (M \ F (D(2ε)))× [0, 1] a product Riemannian metric under the map Φ – notethat F (D(ε)) and W (2ε) have disjoint closures. The vector bundle E on W whenpulled back to (M \ F (D(ε))) × [0, 1] is necessarily isomorphic to the pull-backof a bundle on (M \ F (D(ε))) under the cartesian projection on (M \ F (D(ε))).We assume the hermitian inner product to be the pull-back of one on the bundleon (M \ F (D(ε))). We note that the complement of W (3ε) in W is contained inF (D(4ε)) and hence E is trivial over this complement. Observe that, if ν denotesthe unit inward normal field along the boundary, the interior product with ν yieldsan isomorphism from the bundle ΩW ofm+1-forms onW restricted toM (resp.M ′)to ΩM (resp. Ω′

M ), the bundle of m-forms on M (resp. M ′), which in additionis compatible with the flat connections on these bundles. Now consder the closedΩM -valued m-form TE on M and let α denote the pull-back of its restriction toM \ F (D(3ε)) to W (3ε) (identified with (M \ F (D(3ε))) × [0, 1] via Φ). Then αrestricted to Φ((M \ F (D(3ε)))× 1 = M ′ \ F (D(3ε)) is the same as the restrictionof T ′

E (on M ′). Let α be a m-form on W which equals α on W (3ε), TE on M andT ′E on M ′.

Applying Stokes’ theorem to the ΩW -valued m form α, we get∫M ′

α−∫M

α =

∫W

dα.

Now since the form α is closed, so is its pull back to W (3ε).) It follows that theintegral on the right hand side equals the integral over F (D(4ε)). Now there is anelementary cobordism V between the sphere Sm and Sr × Ss admitting a Morsefunction g : V → [−1, 1] with exactly one non-degenerate critical point of index radmitting a neighbourhood with properties entirely analogous to the neighbourhood

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12 M.S.RAGHUNATHAN

U of w in W . Let F (resp. F , F ′) be the trivial bundle of rank equal to that ofE. Then we see that the same argument as above shows that the integral overF (D(4ε)) is equal to ∫

Sr×Ss

T ′F −

∫Sm

TF

and this is zero as the analytic index of the signature operator for the trivial bundleon the product of two spheres is zero as can be checked easily (using the Hodge-DeRham theorem). This proves Theorem 8.

7. Proof of the Atiyah - Singer theorem

In this section we complete the proof of the theorem. We first dispose of thecase of odd-dimensional M in the same way as is done in [4]: we show that boththe analytic and topological indices vanish.

7.1. The odd-dimensional case. When M is odd-dimensional the asymp-totic expansion of the heat kernel is in odd powers of t1/2 so that for any vectorbundle E on M (with a unitary connection ω), TE is zero. Hence a(DE) = 0. Onthe other hand t(DE) is zero as the K-theoretic symbol of any elliptic differentialoperator itself is zero. This is seen as follows. As we have seen, we may assume thatthe operator has even degree. But this means that the symbol and hence also theK-theoretic symbol is invariant under the bundle automorphism −(Identity) whilethis automorphism induces the map −(Identity) on K0(B,S) as the antipodal mapon even-dimensional spheres is orientation reversing.

7.2. The case of spheres. When the manifold M is an even-dimensionalsphere, the group K0(B,S) ⊗ Q is isomorphic to Q2 and is generated as a vectorspace over Q by D1 and the de Rham-Hodge operator. Here 1 denotes the trivialline bundle (see [4]). For the first of these two operators, it is easily checked (aswas observed earlier) that both indices vanish; for the second the equality of thetwo operators follows from the Gauss-Bonnet theorem (see [4]). Thus the theoremholds for all spheres.

7.3. Products. If M and M ′ are two manifolds for which the analytic andtopological indices are equal the same holds for M × M ′. This is easily seen asfollows.

(i) K0(M ×M ′)⊗Q is isomorphic to (K0(M)⊗Q)⊗ (K0(M ′)⊗Q) underthe natural map induced by the formation of “external” tensor productsand this is compatible with the Kunneth isomorphism in cohomology andthe Chern character; note also that the isomorphism Th for M × M ′ isthe tensor product of the Th for the two factors. This implies that thetopological index is multiplicative: if E (resp. E′) is a vector bundle onM (resp. M ′) and p (resp. p′) is the Cartesian projection of M ×M ′ onM (resp. M ′), then t(p∗(E)⊗ p′∗(E′)) = t(E).t(E′).

(ii) That a(p∗(E) ⊗ p′∗(E′)) = a(E).a(E′) follows from the fact that thetwisted signature operator on (p∗(E) ⊗ p′∗(E′)) is obtained from the E-and E′-twisted operators on M and M ′ respectively by taking the tensorproduct of these last operators.



Since we have proved the theorem for spheres and on the even-dimensional spherethere is a bundle E with a(E) non-zero – the one which corresponds to the deRham-Hodge operator - it follows that the theorem holds for M if (and only if) itholds for M × S2q.

7.4. Spherical cohomology classes. We need the following result due toSerre (see [5]):

Theorem 7.1. Let X be a finite C-W complex of dimension n and α an elementof Hq(X,Q). Assume that n < 2q−1. Then there is a continuous map f : X → Sq

and an element α0 in Hq(Sq,Q) such that f∗(α0) = α.

7.5. Conclusion of the proof. Since the Chern character is an isomorphismfrom K0(M)⊗Q to Heven(M ;Q), we may view the analytic and topological indicesas homomorphisms from Heven(M ;Q) to Q. Suppose now that the theorem holdsfor all manifolds of (even) dimension less than m, the dimension of M . Let α be anelement of Hq(M ;Q) with q an even integer greater than or equal to zero. We needto prove that a(α) = t(α). If q = 0, this is the Hirzebruch signature theorem (asis shown by an explicit computation of the topological index – the analytic index,it is easy to show, is the signature of the manifold). Assume then that q ≥ 2.Let l be an even integer such that m + q < 2(q + l). Let β be a generator ofH l(Sl;Q) and let γ be the class in Hq+l(M × Sl;Q) which corresponds to α ⊗ βunder the Kunneth isomorphism. It suffices to show that a(γ) = t(γ) since thetheorem holds for spheres, a(β) = t(β) is non-zero and one has a(γ) = a(α).a(β)and t(γ) = t(α).t(β).

Now, in view of Serre’s theorem, there is a smooth map f : M × Sl → Sq+l

such that γ is the pull-back of a (q + l)-cohomology class η of Sq+l under f . NowM×Sl is the boundary of W = M×Dl+1 and we can extend f to a smooth map Fof W into the disc Dq+l+1. Let Z be an interior point of the q+ l+1 disc which isnot a critical value of F . Let U be an open disc containing Z such that the closureof U is disjoint from the set of critical values. Then the inverse image of Dq+l+1 \Uunder F gives a cobordism W ′ between M × Sl and the inverse image N of theboundary of U in Dq+l+1. This last manifold is diffeomorphic to the product of amanifold M ′ of dimension m + l + 1 − (q + l + 1) = m − q < m and a sphere ofdimension q + l. If γ (resp. γ′) denotes the inverse image of η in the cohomologyof W ′ (resp. N), by the bordism invariance theorem a(γ) = a(γ′) and t(γ) = t(γ′).On the other hand, since dimM ′ < dimM ′, the theorem holds for M ′ and hencealso for its product N with Sl. Hence a(γ) = t(γ). The theorem follows.

8. Appendix

Let X be a finite CW-complex of dimension N and W a real vector bundle onX of rank m equipped with an inner product. Let B (resp. S) be the unit disc(resp. sphere) bundle in W . Let ε denote the antipodal map along the fibres of S.Let P denote the real projective space bundle associated to W : P is the same asthe quotient of S obtained by identifying each point s of S with ε(s). Let p (resp.q) be the projection of S (resp. P ) on M and u the natural map from S to P sothat q · u = p. Let E and F be complex vector bundles on X of rank n and ES

and FS (resp. EP and FP ), their pull-backs to S (resp. P ). Let σt, t ∈ [0, 1], be a

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14 M.S.RAGHUNATHAN

homotopy of sections of the bundle Iso(ES , FS) of isomorphisms between fibres ofES and FS over S. With this notation we have the following.

Proposition 8.1. Assume that σ0 and σ1 are lifts of sections τ0 and τ1 ofIso(EP , FP ). In other words σ0 and σ1 are invariant under the antipodal map ε ofS. Let ν(n) = 2(m−1).n. Then there is a family θ(t,s), (t, s) ∈ [0, 1]×[0, 1] of sectionsof Iso(ν(n).ES, ν(n).FS) depending continuously on (t, s) such that θ(t,0) = ν.σt,θ(0,s) and θ(1,s) are independent of s, and θ(t,1) is ε-invariant.

We argue by induction on the dimension ofX. When dimX = 0, the assertion isobvious. Assume then that it is proved for all finite complexes of dimension less thann = dimX. Let Y denote the (n−1)-skeleton of X; then X is obtained by attachingthe disjoint union A =

∐1≤i≤r Di of n-discs through a map φ of its boundary ∂A =∐

1≤i≤r ∂Di in Y . Fix trivialisations of the pull-backs of W , E and F to A. We may

then regard the pull-backs of σt (resp. τ0, τ1) to A as GL(N,C)-valued functionson A×Sm−1 (resp. ∂A×Pm−1) which we denote by ft (resp. g0, g1) (here P

m−1 isthe (m− 1)-dimensional real projective space). By the induction hypothesis, thereis a family ′σ(t, s) of sections of Iso(ES , FS) over the inverse image SY of Y in Swith ′σ(t,0) = ν(n − 1).σt,

′σ(0,s) = ν(n − 1).σ0,′σ(1,s) = ν(n − 1).σ1 and ′σ(t,1)

invariant under ε. This means that there is a family ′f(t,s) of maps from ∂A×S toGL(ν(n− 1)N,C) such that ′f(t,0) = ν(n − 1).ft,

′f(t,1) is ε-invariant, ′f(0,s) = g0and ′f(1,s) = g1. Now using the homotopy extension property, one sees that there isa family F(t,s) of maps from A×S to GL(ν(n−1)N,C) such that F(t,s) =

′f(t,s) on∂A×S, F(0,s) = f0 and F(1,s) = f1. Let ht = F(t,1). Then for each i with 1 ≤ i ≤ r,

ht, t ∈ [0, 1], defines a map from the boundary of Di ×Pm−1 × [0, 1] (which can beidentified with Sn+1×Pm−1) to GL(ν(n−1)N,C). Equivalently we have a map Hfrom Pm−1 to Ωp(GL(ν(n− 1)N,C), the p-th free loop space of GL(ν(n− 1)N,C),which when composed with the projection u of Sm−1 on Pm−1 factors throughthe space of all maps from Dn+1 to GL(ν(n − 1)N,C). Now homotopy classesof maps from Pm−1 to Ωp(GL(ν(n − 1)N,C)) can be identified with the groupK−p−1(Pm−1) so that H defines an element [H] in that group. Moreover theimage of [H] in K−p−1(Sm−1) under the map u∗ induced by u is trivial. Sincethe kernel of the map u∗ : K−p−1(Pm−1) → K−p−1(Sm−1) is annihilated by 2m−1

(this is seen easily as a consequence of the Atiyah-Hirzebruch spectral sequence forK-theory [1] and the fact that the kernel of the cohomology map from Pm−1 toSm−1 is a direct sum of m− 2 copies of Z/2), the proposition follows.

References

[1] M.F.Atiyah and F.Hirzebruch, Vector bundles and homogeneous spaces, 1961 Proc. Sympos.Pure Math., Vol. III pp 7–38, American Mathematical Society, Providence, RI.

[2] M.F.Atiyah and I.M.Singer, The index of elliptic operators on compact manifolds, Bull. Amer.Mah. Soc. 69 (1963), 422–433.

[3] S.Minakshisundaram and A.Pleijel, Some properties of the eigenfunctions of the Laplace-

operator on Riemannian manifolds, Canadian J. Math. 1 (1949), 242–256.[4] Richard S.Palais, Seminar on the Atiyah-Singer index theorem, Annals of Mathematics Stud-

ies, No. 57, Princeton University Press, Princeton N. J. 1965.[5] J.-P.Serre, Groupes d’homotopie et classes des groupes abeliens, Ann. of Math. (2) 58 (1953),

258–294.






Spin(7) Instantons and the Hodge Conjecture for CertainAbelian Four-folds: a Modest Proposal

T. R. Ramadas

Dedicated to S. Ramanan on the occasion of his seventieth birthday.

Abstract. The Hodge Conjecture is equivalent to a statement about condi-tions under which a complex vector bundle on a smooth complex projectivevariety admits a holomorphic structure. In the case of abelian four-folds, recentwork in gauge theory suggests an approach using Spin(7) instantons. I adver-

tise a class of examples due to Mumford where this approach could be tested.I construct explicit smooth vector bundles - which can in fact be constructedin terms of of smooth line bundles - whose Chern characters are given Hodgeclasses. An instanton connection on these vector bundles would endow themwith a holomorphic structure and thus prove that these classes are algebraic.I use complex multiplication to exhibit Cayley cycles representing the givenHodge classes. I find alternate complex structures with respect to which thegiven bundles are holomorphic, and close with a suggestion (due to G. Tian)as to how this may possibly be put to use.

1. Introduction

Let X be a smooth complex projective variety of dimension n, and c a rational(p, p) cohomology class (0 < p < n). The Hodge Conjecture is that

H: there exist finitely many (reduced, irreducible) (n− p)-dimensional sub-varieties Yi and rational numbers ai such that c =

∑i ai[Yi], where [Yi] is

the (rational) cohomology class dual to Yi . That is, c is dual to a rationalalgebraic cycle.

This is equivalent to

V: there exists a holomorphic vector bundle E such that its Chern characterch(E) is equal to a rational multiple of c modulo (classes of) rationalalgebraic cycles.

The second statement implies the first because the Chern character of a holo-morphic (and therefore algebraic) bundle factors through the Chow ring of algebraic

2010 Mathematics Subject Classification. 32J25, 14C30, 70S15.It is a pleasure to thank Bobby Acharya, who told me of Spin(7) instantons and gave me

a copy of Christopher Lewis’ thesis. I am also grateful to Dominic Joyce, Nigel Hitchin, M.S.Narasimhan, and Gang Tian for very helpful comments.


1



155

2 T. R. RAMADAS

varieties. The converse also holds. In fact, as Narasimhan pointed out to me, it isknown ([M]) that the rational Chow ring is generated by stable vector bundles.

Let X, c be as above. By a theorem of Atiyah-Hirzebruch ([A-H], page 19),the Chern character map ch : K0(X) ⊗ Q → Heven(X,Q) is a bijection, whereK0(X) is the Grothendieck group of (topological/smooth) vector bundles on X.Thus we are assured of the existence of a smooth bundle E and and an integern > 0 such that ch(E) = rank(E) + nc. A possible strategy to show that a givenclass c is algebraic suggests itself – find a suitable such bundle E and then exhibita holomorphic structure on it. This note is written to argue that recent progressin mathematical gauge theory, and in particular the work of G. Tian and C. Lewis,makes this worth pursuing, at least in the case of certain abelian four-folds. Suchan approach to the Hodge Conjecture for the case of Calabi-Yau four-folds is surelyknown to the experts (and this has been confirmed to me), but I have only beenable to locate some coy references. Claire Voisin ([V]), following a similar approach,has much more definitive negative results in the case of non-algebraic tori.

Before proceeding, let us note that the known “easy” cases of the Hodge con-jecture are proved essentially by the above method. First, given an integral classc ∈ H2(X,Z), a smooth hermitian line bundle L exists with (first) Chern class equalto c. Given any real 2-form Ω representing c there exists a unitary connection onL with curvature −2πiΩ. If c is a (1, 1) class, it can be represented by an Ω whichis (1, 1). The corresponding connection defines a holomorphic structure on L. If cis an integral (n − 1, n − 1) class, the strong Lefschetz theorem exhibits the dualclass as a rational linear combination of complete intersections.

What follows is the result of much trial and error and computations - which Ieither only sketch or omit altogether - using Mathematica; the notebooks are avail-able on request. (I used an exterior algebra package of Sotirios Bonanos, availablefrom http://www.inp.demokritos.gr/~sbonano/. )

2. Mumford’s examples

We consider Hodge classes on certain abelian four-folds. These examples aredue to Mumford ([P]).

It is best to start with some preliminary algebraic number theory. If F is analgebraic number field, with degree F = d, the ring of algebraic integers Λ ≡ oF isa free Z-module of rank d which generates F as a Q-vector space. If V denotes thereal vector space R⊗Q F , then Λ ⊂ V is a lattice and Xr = V/Λ is a real d-torus.

Let L denote the Galois saturation of F in Q ⊂ C. (That is, L is the smallestsubfield Galois over Q and containing any (and therefore all) embeddings of F .)Then G = Gal(L/Q) acts transitively on the set E of embeddings ι : L ↪→ Q by(g, ι) �→ g(ι) = g ◦ ι (g ∈ G, ι ∈ E), and the image by ι is the fixed field of thestabiliser of ι. Further, the map

Q⊗Q F → QE

given by 1⊗ x �→ (ι(x))E is an isomorphism of Q vector spaces.Turning to the real torus Xr:

(1) we have natural isomorphisms H1(Xr,Z) = Λ and H1(Xr, Q) = QE ;

(2) H1(Xr, Q) has basis {dtι}E , where dtι is induced by the projection to theιth factor from QE .

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INSTANTONS AND THE HODGE CONJECTURE 3

In what follows we will identify the real or complex cohomology of Xr with thecorresponding spaces of translation-invariant forms on Xr.

We will need the following result, whose proof is straightforward.

Proposition 2.1. A one-form ω =∑

ι ωιdtι represents a rational class iff thecoefficients ωι belong to L and satisfy the equivariance

ωg(ι) = g(ωι), g ∈ G

Similarly, a two-form φ =∑

ι,κ φι,κdtι ∧ dtκ (with the coefficients antisymmetric

functions of the two indices) represents a rational class iff

φg(ι),g(κ) = g(φι,κ), g ∈ G

Suppose now that the embeddings E occur in complex conjugate pairs - E =E′ E′′, with each ι ∈ E′ corresponding to ι ∈ E′′. Then the map

V = R⊗Q F (↪→ C⊗Q F ∼ CE) → C

E′

is an isomorphism of real vector spaces and induces a (translation-invariant) com-plex structure on Xr, which becomes a complex torus, which we will denote simplyX.

We turn now to specifics. Let P = ax4+bx2+cx+d be an irreducible polynomialwith rational coefficients and all roots x1, x2, x3, x4 real. We will suppose that theroots are numbered such that x1 > x2 > x3 > x4. Let L1/Q be the splitting fieldL1 = Q[x1, x2, x3, x4] ⊂ R. We suppose that P is chosen such that the Galoisgroup is S4. This is equivalent to demanding that [L1 : Q] = 24. We set L ≡ L1[i].This is a Galois extension of Q, with Galois group S4 × {e, ρ}, where ρ is complexconjugation.

Consider a cube, with vertices labeled as in the figure:

1

23

4

2b 3b

4b1b

Let G denote the group of symmetries of the cube. We have the exact sequence:

1 → {e, ρ} → G → S4 → 1

157

4 T. R. RAMADAS

where now ρ denotes inversion, and S4 is the group of permutations of the fourdiagonals. Splitting this, identifying S4 with (special orthogonal) rotations imple-menting the corresponding permutation of diagonals. we get an identification

G ∼ S4 × {e, ρ} = Gal(L/Q)

Let H denote the stabiliser of the vertex 1, F the corresponding fixed field, andϕ1 : F → L → C the corresponding embedding. The left cosets of H can beidentified with the vertices of the cube, as well as embeddings of F in C. We labelthe latter ϕj , ϕj (j = 1, 2, 3, 4).

Note that the field F is invariant under complex conjugation, which thereforeacts on it with fixed field F1. Clearly, F1 = Q[x1] . We set

D = (x1 − x2)(x1 − x3)(x1 − x4)(x2 − x3)(x2 − x4)(x3 − x4)

Given our ordering of the roots, D > 0. Note that iD ∈ F , F = F1[iD], and Δ ≡ D2

is a rational number. We will assume that (after multiplying all the xi by a commonnatural number if necessary) Δ is an integer (and so D is an algebraic integer). Wewill repeatedly use the fact that the Galois conjugates of iD ∈ F are given by

φj(iD) = −(−1)jiD

φj(iD) = (−1)jiD(2.1)

In our case Xr is a real 8-torus. The embeddings ϕi : F → C induce R-linearmaps zi : V → C, such that z = (z1, z2, z3, z4) is an isomorphism of R-vector spacesV → C4. We let X denote the complex manifold V/Λ obtained thus. Note thatif a ∈ oF , multiplication by a is a Q-linear map F → F which induces a R-linearmap V → V taking the lattice Λ to itself. If z(a) = (a1, a2, a3, a4), and u ∈ V withz(u) = (z1, z2, z3, z4) we also have z(au) = (a1z1, a2z2, a3z3, a4z4), so that we seethat this induces an analytic map (in fact an isogeny) X → X. In other words, oFacts on X by “complex multiplication”.

As a complex torus, X is certainly Kahler, and we shall see below that itis algebraic. What is relevant for our purposes is that it is possible to describeexplicitly the Hodge decomposition as well as the rational structure of the complexcohomology of X. Let T (for “top”) denote the set of indices {1, 2, 3, 4} and B (for“bottom”) the indices {1, 2, 3, 4}. (The corresponding vertices are denoted 1b, etc.in the figure.)

Proposition 2.2. A basis of Hp,q is labeled by subsets P ⊂ T , Q ⊂ B, with|P | = p, and |Q| = q, and given by the translation-invariant forms dzP dzQ, wherefor example, if P = {i, j}, with i < j we set dzP = dzi ∧ dzj , and if Q = {i, j}(again with i < j), we set dzQ = dzi ∧ dzj. A basis of the rational cohomology Hr

Q

is labelled by pairs (R,χ) where

• R is an orbit of G in the set of sequences μ ≡ (μ1, . . . , μr) of distinctelements in T ∪B, and

• χ runs over a Q-basis of HR, the space of G-equivariant maps R → L,satisfying

χ(μσ(1), . . . , μσ(r)) = sign(σ)χ(μ1, . . . , μr)

for any permutation σ such that μ, μσ ∈ R.

158


The corresponding classes are given by the forms∑μ∈R

χ(μ)dzμ

We use the notation dzμ = dzμ1∧· · ·∧dzμr

, with the convention that dz1 = dz1, etc.

It is useful to note the following

Lemma 2.3. Given R, the Q-dimension of HR is |R|/r!.

Note that if r = 2p, a rational class as above is of type (p, p) iff the orbit consistsof sequences with elements equally divided between the top and bottom faces ofthe cube. In particular, the rational (1, 1) classes correspond to the G-orbit of thesequence (1, 1). Since in this case HR has dimension 4, we see that the Neron-Severigroup has rank 4.

Consider now the orbit of the sequence (1, 3, 2, 4). This corresponds to a two-dimensional space M of rational (2, 2) classes, which have the property that theseare not products of rational (1, 1) classes. It is easy to check that but for (theQ-span of) these, rational (2, 2) classes are generated by products of rational (1, 1)classes.

Proposition 2.4. A Q-basis of M is given by the classes

• M = D(dz1dz2dz3dz4 + dz1dz2dz3dz4)• M ′ = i(dz1dz2dz3dz4 − dz1dz2dz3dz4)

So the Hodge conjecture in this case would be that : the classes M and M ′ arealgebraic.

We will use complex multiplication in an essential way later; here I illustrate itsuse by showing how it can be used to halve our work. Consider multiplication bythe algebraic integer a = 1 + iD ∈ oF . This induces a (covering) map πa : X → Xand one easily computes:

π∗aM = ((1−Δ)2 − 4Δ)M + 4(1−Δ)ΔM ′

π∗aM

′ = ((1−Δ)2 − 4Δ)M ′ − 4(1−Δ)M(2.2)

This proves

Proposition 2.5. Algebraicity of either one of M or M ′ implies that of theother.

Before moving on, we find a positive rational (1, 1) form ω on X, which willshow that it is projective. Let μ1 ∈ F1 (to be chosen in a moment) and considerthe form

ω =iD

Δ(μ1dz1dz1 − μ2dz2dz2 + μ3dz3dz3 − μ4dz4dz4)

where μi are Galois conjugates. Clearly this is a rational (1, 1) form, and it will bepositive provided (−1)j+1μj > 0. For example, we can take μ1 = (x1 − x2)(x1 −x3)(x1 − x4), and we will do so. With this choice the holomorphic four-form θ ≡(1/D)dz1dz2dz3dz4 satisfies

(2.3)ω4

4!= θ ∧ θ

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6 T. R. RAMADAS

3. Expressing M, M ′ in terms of Chern characters

Consider the G-orbit of (1, 3). The corresponding subspace of H2Qis spanned

by the classes of the form

A1 = a13(x1 − x3)dz1dz3 + ....

where a13 belongs to the fixed field of the subgroup of G that leaves the set ofvertices {1, 3} invariant, and this coefficient determines the others in the sum byGalois covariance. We introduce the notation

Ta = a13a24(x1−x3)(x2−x4)−a12a34(x1−x2)(x3−x4)+a14a32(x1−x4)(x3−x2)

Squaring A1, we get

A21 =2a13a24(x1 − x3)(x2 − x4)dz1dz3dz2dz4 + ..

+2a12a13(x1 − x2)(x1 − x3)dz1dz2dz1dz3 + ...

+2a12a21(x1 − x2)(x2 − x1)dz1dz2dz2dz1 + ..

+2Tadz1dz3dz2dz4 + ..

If we make the replacement a13 � icDa13 (c an integer introduced for later use in§7), we get a class A2, such that

A22/(c

2Δ) =2a13a24(x1 − x3)(x2 − x4)dz1dz3dz2dz4 + ..

+2a12a13(x1 − x2)(x1 − x3)dz1dz2dz1dz3 + ...

+2a12a21(x1 − x2)(x2 − x1)dz1dz2dz2dz1

−2Tadz1dz3dz2dz4 − ..

Suppose now that the classes Ai are integral. (This is easily arranged by clear-ing denominators.) Let Li (i = 1, 2) be the line bundle with Chern class Ai.

Proposition 3.1. Let Vi = Li ⊕ L−1i , i = 1, 2. Then

ch(Vc2Δ1 � V2) = 4c2Δ(Tadz1dz3dz2dz4 + ..)

where the equality is modulo (rational) 0- and 8-forms.

We have the freedom to choose the coefficient a13, which by Galois covariancedetermines the other coefficients, and hence the above classes. We now make thechoice

a13 = h3

where for later use we introduce the notation

h2 = (x1x2 + x3x4)

h3 = (x1x3 + x2x4)

h4 = (x1x4 + x2x3)

(3.1)

Then Ta = −D, and we get

Theorem 3.2. With the above choice,

ch(Vc2Δ1 � V2) = 4c2ΔM

where the equality is modulo (rational) 0- and 8-forms.

160


The virtual bundle Vc2Δ1 � V2 has the properties: c1 = 0, and c2 ∧ ω2 = 0,

where ω is the rational Kahler class defined at the end of §2. (This is because theMi, as can be easily seen, are orthogonal to ω.) This will not do for reasons to dowith the Bogomolov inequality, but this can be fixed because of a minor miracle:

Proposition 3.3. With the above choices,

A21 ∧ ω = −2iΔ

1

μ4dz1dz1dz2dz2dz3dz3 + ...

In particular, A21 ∧ ω is a (rational) (3,3) form.

For later use, we also record

Proposition 3.4. With the above choices,

A1 ∧ ω3 = 0

A2 ∧ ω3 = 0

We will suppose that kω (for some positive integer k) is an integral class, andlet Lkω denote a (holomorphic, in fact ample) line bundle with this Chern class.The following is a easy consequence of 3.3.

Theorem 3.5. Let V1 = L1 ⊗ Lkω ⊕ L−11 ⊗ Lkω, and V2 = L2 ⊕ L−1

2 and set

E = Vc2Δ1 � V2. Then

ch(E) = 2c2Δkω + 4c2ΔM + k2c2Δω2

where the equality is modulo (rational) 0-, (3,3)- and 8-forms.

In particular, this (difference) bundle E satisfies the “Bogomolov inequality”:

< c2ω2 > − 2Δ− 3

4(Δ− 1)< c21ω

2 > =c2Δ

c2Δ− 1k2 < ω4 >

> 0

The symbol < .. > stands for integration against the fundamental class. We usethe quote marks since we are not (yet!) talking of a holomorphic bundle E . Sincethe virtual bundle has positive rank, we are justified, up to some non-canonicalchoices, in dropping the qualifiers “virtual”/“difference”.

Remark 3.6. We have concentrated on the Hodge class M in this section; itis possible, with slight modifications to the above expressions, to find a smoothbundle E ′ whose Chern character similarly contains the Hodge class M ′.

4. Spin(7) instantons

In this section we recall the definition of Spin(7) instantons ([B-K-S], [T]),specialised to the case of a Kahler four-fold X with trivial canonical bundle KX .We fix a Ricci-flat Kahler form ω, and let θ denote a trivialisation of KX satisfying(2.3). We define a (complex antilinear) endomorphism � : Ω(0,2) → Ω(0,2), by

|α|2θ = α ∧ �α

We have �2 = 1, so we can decompose the bundle into a self-dual and anti-self-dualpart:

Ω(0,2) = Ω(0,2)+ ⊕ Ω

(0,2)−

161

8 T. R. RAMADAS

Let E be a hermitian (C∞) vector bundle on X. A Spin(7) instanton is ahermitian connection A on E, whose curvature F satisfies

F(0,2)+ = 0, ΛF = 0

Here Λ denotes as usual contraction with the Kahler form. A crucial point is thefollowing ([T],[L]):

Proposition 4.1. The L2-norm of the curvature of a Spin(7) instanton satis-

fies ||F (0,2)− ||22 =

∫Tr(F ∧ F ) ∧ θ

In particular, if the invariant on the right vanishes, a Spin(7) instanton isequivalent to a holomorphic structure on E together with a Hermite-Einstein con-nection. Clearly, such a bundle would be poly-stable, and hence (or directly fromthe Hermite-Einstein condition) satisfy the Bogomolov inequality:

(4.1) c2(E).ω2 ≥ r − 1

2rc1(E)2.ω2

where r denotes the rank of E.Now that we have embedded the problem of construction a holomorphic structure

on E in a broader context – that of constructing an instanton connection – one canenvisage deforming the complex structure in such a way that∫

c2(E) ∧ θ �= 0

and still hope to have the moduli space of semi-stable holomorphic structures on Edeform as the moduli space of instanton connections.

There are several possible approaches to the construction of such a connection.

(1) Exhibit an instanton by glueing.(2) The fact that the bundles are exhibited as a difference of two vector bun-

dles, each of which is in turn a sum of explicit line bundles, suggests theuse of monads, possibly combined with a twistor construction. This wouldinvolve a matrix of sections of line bundles.

A third idea, suggested to me by G. Tian, is pursued in the last section of thispaper.

5. Calibrations; Cayley submanifolds

In his thesis, C. Lewis [L] shows how (in one particular case) one can constructan instanton by glueing around a suitable Cayley submanifold. (See also [B].) Wedefine these terms below, and then exhibit some relevant Cayley cycles that arisein our context. (References are [H-L], and [J]; but we follow the conventions of[T].)

Definition 5.1. Let M be a Riemannian manifold. A closed l-form φ is saidto be a calibration if for every oriented tangent l-plane ξ, we have

φ|ξ ≤ volξ

where volξ is the (Riemannian) volume form. Given a calibration φ, an orientedsubmanifold N is said to be calibrated if φ restricts to N as the Riemannian volumeform.

162


It is easy to see that a calibrated submanifold is minimal. Two examples arerelevant. First, if M is Kahler, with Kahler form ω, for any integer p ≥ 1, theform ωp

p! is a calibration, and the calibrated submanifolds are precisely the complex

submanifolds.The case that concerns us is that of a four-fold X with trivial canonical bundle

KX . We fix an integral Ricci-flat Kahler form ω, and let θ denote a trivialisationof KX with normalisation as in (2.3). Then 4Re(θ) is a second calibration, andthe calibrated submanifolds are called Special Lagrangian submanifolds. There is a“linear combination” of the two, defined by the form

Ω =w2

2+ 4Re(θ)

which defines the Cayley calibration. The corresponding calibrated manifolds arecalled Cayley manifolds. Any smooth complex surface (on which the second termwill restrict to zero) or any Special Lagrangian submanifold (on which the first termwill vanish) furnish examples. In fact, the Cayley cycles we deal with will be of thelatter kind.

Cayley manifolds are not easy to find. We will use the following result (Propo-sition 8.4.8 of [J]):

Proposition 5.2. Let X be as above, and σ : X → X an anti-holomorphicisometric involution such that σ∗θ = θ. Then the fixed point set is a Special La-grangian submanifold.

We return to the constructions of our paper. Recall that the field F is invariantunder complex conjugation, which therefore acts on it with fixed field F1. Thisinduces an involution σ1 : V → V such that z(σ1(u)) = z(u), where, if z =(z1, z2, z3, z4), we set z = (z1, z2, z3, z4). The induced involution σ1 : X → X hasfixed locus which we will denote Y . Note that σ satisfies the conditions of theprevious Proposition and therefore Y is Special Lagrangian.

Theorem 5.3. There exist (rational) Cayley cycles representing the Hodgeclasses Mi.

Proof. Recall the isogeny πa : X → X, given by multiplication by the alge-braic integer a = 1 + iD. It is easy to check

π∗aω = (1 +Δ)ω

π∗aθ = (1 +Δ)2θ

We will also need a second isogeny πb, where b = iD, which satisfies

π∗bω = Δω

π∗b θ = Δ2θ

These equations guarantee the maps πa, πb take Cayley cycles to Cayley cycles(possibly introducing singularities.)

We have the following table giving the action of the above isogenies on four-forms of various types (all the forms in the list are eigenvectors):

163

10 T. R. RAMADAS

Form eigenvalue of π∗a eigenvalue of π∗

b “multiplicity”dz1dz2dz3dz4 (1 + Δ)2 Δ2 2× 1dz1dz1dz2dz3 (1 + Δ)2 Δ2 2× 8dz1dz1dz2dz4 (1 + Δ)(1− iD)2 −Δ2 2× 4dz1dz2dz3dz4 (1 + Δ)(1− iD)2 −Δ2 2× 4dz1dz1dz2dz2 (1 + Δ)2 Δ2 6dz1dz1dz2dz3 (1 + Δ)(1− iD)2 −Δ2 2× 12dz1dz2dz3dz4 (1 + Δ)2 Δ2 4dz1dz2dz3dz4 (1 + iD)4 Δ2 1dz1dz2dz3dz4 (1− iD)4 Δ2 1

(We list only forms of type (4,0), (3,1) and (2,2), omitting types that are related tothe ones in the list by conjugation. The term “multiplicity” refers to the numberof forms of a given type, not the multiplicity of eigenvalues.)

Consider the operator

Φa = (π∗a − (1 + Δ)2)(π∗

b +Δ2)

From the list it follows that the space M⊗Q C (spanned by the Mi) is the sum ofthe eigenspaces of Φa corresponding to the non-zero eigenvalues. We have (using(2.2))

Φ∗aM

′ = −8Δ2[2ΔM ′ + (1−Δ)M ]

Φ∗aM = −8Δ2[−(1−Δ)ΔM ′ + 2ΔM ]

Next, note that the Cayley cycle Y defined above satisfies

< Y,M > = 2Dδ

< Y,M ′ > = 0

Here <,> denotes the integration pairing of cycles and forms, and δ denotes theco-volume of the lattice oF1

⊂ F1 ⊗Q R. By standard facts in algebraic numbertheory, δ is a rational multiple of D; so the above pairings are rational, as they hadbetter be.

We now consider the Cayley cycle

Ca = (πa − (1 + Δ)2)(πb +Δ2)Y

By construction Ca is orthogonal to all the forms in the above list except the Mi.Its pairings with these are as follows:

< Ca,M > = −32Δ3Dδ

< Ca,M′ > = −16Δ2(1−Δ)Dδ

Let now a = (1 − iD), and repeat the above construction with operators Φa,etc.

Φ∗aM

′ = −8Δ2[2ΔM ′ − (1−Δ)M ]

Φ∗aM = −8Δ2[(1−Δ)ΔM ′ + 2ΔM ]

This gives a cycle Ca satisfying

< Ca,M > = −32Δ3Dδ

< Ca,M′ > = 16Δ2(1−Δ)Dδ

Clearly the theorem is proved. �

164


Remark 5.4. The above result, though suggestive, does not take us far. This isbecause the above “Cayley cycle” is not effective, but in fact a linear combination ofSL subvarieties with both positive and negative coefficients. (D. Joyce has pointedout that this must be the case given that it represents a (2, 2) class.) To makematters worse, a theorem of G. Tian (Theorem 4.3.3 of [T]) states that blow-uploci of Hermite-Yang-Mills connections are effective holomorphic integral cyclesconsisting of complex subvarieties of codimension two. So any glueing will call forvery new techniques.

6. Adapted complex structures

In this section we seek translation-invariant complex structures on the eight-torus V/Λ such that the classes Ai are of type (1, 1) w.r.to these complex structures,and therefore define holomorphic structures on the line bundles Li. The originalmotivation was to exploit twistor techniques for the construction of instantons, butwe postpone discussion of possible uses of this investigation to the last section.

Consider a linear change of coordinates of the form

z1 = w1 + α12w2 + α14w4

z3 = w3 + α32w2 + α34w4

z2 = w2 + α21w1 + α23w3

z4 = w4 + α41w1 + α43w3

We collect the coefficients into 2× 2 matrices α and α as follows:

α =

(α12 α14

α32 α34

)

and

α =

(α21 α23

α41 α43

)

and rewrite the above change of coordinates as follows:(z1z3

)=

(w1

w3

)+ α

(w2

w4

)(z2z4

)=

(w2

w4

)+ ¯α

(w1

w3

)

A long but straightforward computation shows Ai will be of type (1, 1) provided:

h3(x1 − x3)(α12α34 − α14α32)

+h4(x1 − x4)α12 − h2(x1 − x2)α14

+h2(x3 − x4)α32 − h4(x3 − x2)α34

+h3(x2 − x4) = 0

and

h3(x2 − x4)(α21α43 − α23α41)

+h4(x1 − x4)α43 − h2(x3 − x4)α41

+h2(x1 − x2)α23 − h4(x3 − x2)α21

+h3(x1 − x3) = 0

165

12 T. R. RAMADAS

To rewrite these conditions in a more compact form, we introduce some nota-tion:

(1) Given a 2× 2 matrix A:

A =

(a11 a12a21 a22

)

let

(6.1) A =

(a22 −a12−a21 a11

)

(If A is nonsingular, A = (det A)A−1.)(2) Define the symmetric bilinear form <,> on the space of 2× 2 matrices

< A,B >= Tr(AB) = det (A+B)− det A− det B

(3) Let

H =

(−h4(x2 − x3) h2(x3 − x4)−h2(x1 − x2) −h4(x1 − x4)

)

so that

H =

(−h4(x1 − x4) −h2(x3 − x4)h2(x1 − x2) −h4(x2 − x3)

)

The conditions on α and α can now be rewritten:

(6.2) < α,H >= h3(x2 − x4) + h3(x1 − x3)det α

and

(6.3) < α, H >= h3(x1 − x3) + h3(x2 − x4)det α

We assume that the inverse coordinate transformation is of the form(w1

w3

)= c

(z1z3

)+ β

(z2z4

)(w2

w4

)= c

(z2z4

)+ ¯β

(z1z3

)

where c, c are scalars (this will constrain α and α, see below) and β and β 2 × 2matrices. One checks that we then need

c(1− αα) = 1

c(1− ¯αα) = 1

so that we are requiring that αα and ¯αα are scalars. Further,

β = −cα

β = −cα

Note that either αα = ¯αα = 0 and c = c = 1 or

αα =c− 1

c

¯αα =c− 1

c

and c = c. Note also that once α is chosen to satisfy the equation (6.2), then (6.3)is satisfied if we take

α =x1 − x3

x2 − x4

ˆα

166


From now on we will proceed to define α by the above equation. This forces c tosatisfy

c(1− x1 − x3

x2 − x4det α) = 1

Clearly, a necessary condition is

(6.4) det α �= x2 − x4

x1 − x3

We can write down the corresponding almost complex structure. With anobvious schematic notation,

J

(dz1dz3

)= i(2c− 1)

(dz1dz3

)+ 2iβ

(dz2dz4

)

J

(dz2dz4

)= i(2c− 1)

(dz2dz4

)+ 2i

¯β

(dz1dz3

)

By further restricting α one can ensure that ω remains of type (1, 1). Wesummarise our results in

Theorem 6.1. Let the co-ordinates w be defined by(z1z3

)=

(w1

w3

)+ α

(w2

w4

)(z2z4

)=

(w2

w4

)+ ¯α

(w1

w3

)

where the matrix α satisfies

(6.5) < α,H >= h3(x2 − x4) + h3(x1 − x3)det α

and

α =x1 − x3

x2 − x4

ˆα

(α is defined as in (6.1).) Then the forms Ai are of type (1, 1) w.r.to the wi.Further, if α satisfies

α34 = +x1 − x4

x2 − x3α12

α14 = −x3 − x4

x1 − x3α32

(6.6)

then ω remains of type (1, 1).

If α satisfies (6.6), the condition (6.5) becomes

h3(x1 − x3))(x1 − x4

x2 − x3|α12|2 +

x3 − x4

x1 − x2|α32|2) + h3(x2 − x4)

+h4(x1 − x4)(α12 + α12) + h2(x3 − x4)(α32 + α32) = 0(6.7)

The space of solutions J is clearly an 3-dimensional ellipsoid in the two-dimensionalcomplex vector space with co-ordinates (α12, α32). The condition (6.4) becomes:

x1 − x4

x2 − x3|α12|2 +

x3 − x4

x1 − x2|α32|2 �= 0

which corresponds to removing the affine hyperplane H given by

h3(x2 − x4) + h4(x1 − x4)(α12 + α12) + h2(x3 − x4)(α32 + α32) = 0

167

14 T. R. RAMADAS

We have therefore to consider J = J \H, which is the union of two open three-discs.A particular choice of α has remarkable properties. Let

α∗ =1

x1 − x3

((x2 − x3)(1− 2h4

h3) −(x3 − x4)(1− 2h2

h3)

(x1 − x2)(1− 2h2

h3) (x1 − x4)(1− 2h4

h3)

)

Theorem 6.2. With this choice, we have

(x1x3 + x2x4)2

4A1 =(x1 − x2)

2(x2 − x3)2(x1 − x4)

2(x3 − x4)dw1dw2

−(x1 − x2)2(x2 − x3)

2(x1 − x4)(x3 − x4)2dw2dw3

+(x1 − x2)(x2 − x3)2(x1 − x4)

2(x3 − x4)2dw3dw4

+(x1 − x2)2(x2 − x3)(x1 − x4)

2(x3 − x4)2dw4dw1

+(x1 − x2)2(x2 − x3)

2(x1 − x4)2(x3 − x4)dw1dw2

−(x1 − x2)2(x2 − x3)

2(x1 − x4)(x3 − x4)2dw2dw3

+(x1 − x2)(x2 − x3)2(x1 − x4)

2(x3 − x4)2dw3dw4

+(x1 − x2)2(x2 − x3)(x1 − x4)

2(x3 − x4)2dw4dw1

(6.8)

(x1x3 + x2x4)2

4iDA2 =(x1 − x2)

2(x2 − x3)2(x1 − x4)

2(x3 − x4)dw1dw2

+(x1 − x2)2(x2 − x3)

2(x1 − x4)(x3 − x4)2dw2dw3

+(x1 − x2)(x2 − x3)2(x1 − x4)

2(x3 − x4)2dw3dw4

−(x1 − x2)2(x2 − x3)(x1 − x4)

2(x3 − x4)2dw4dw1

−(x1 − x2)2(x2 − x3)

2(x1 − x4)2(x3 − x4)dw1dw2

−(x1 − x2)2(x2 − x3)

2(x1 − x4)(x3 − x4)2dw2dw3

−(x1 − x2)(x2 − x3)2(x1 − x4)

2(x3 − x4)2dw3dw4

+(x1 − x2)2(x2 − x3)(x1 − x4)

2(x3 − x4)2dw4dw1

(6.9)

Δ(x1x3 + x2x4)2

4iDω

= −{(x1 − x2)2(x1 − x3)(x2 − x3)(x1 − x4)

2(x3 − x4)dw1dw1

+(x1 − x2)2(x2 − x3)

2(x1 − x4)(x2 − x4)(x3 − x4)dw2dw2

+(x1 − x2)(x1 − x3)(x2 − x3)2(x1 − x4)(x3 − x4)

2dw3dw3

+(x1 − x2)(x2 − x3)(x1 − x4)2(x2 − x4)(x3 − x4)

2dw4dw4}

(6.10)

In particular, −ω is a Kahler form and the corresponding complex structure makesXr an abelian variety.

Remark 6.3. It is convenient to consider the consider the conjugate complexstructure (w.r.to which holomorphic co-ordinates are the wi. This has the propertythat the forms Ai and ω are of type (1,1), and in addition, ω is Kahler. We let X ′

denote the corresponding abelian variety.

168


7. A strategy

Attempts to invoke twistor methods have not been successful so far. For exam-ple, N. Hitchin pointed out that results of M. Verbitsky make hyperkahler twistorspaces quite unsuitable. G. Tian made the following suggestion: construct instan-tons by deformation (using, say, the continuity method) from a situation whenthey are known to exist. In fact, the complex structure described in the Remark6.3 provides such a starting point. I close with a brief justification for this claim.

With respect to the above complex structure, the bundles Vi defined in thestatement of Theorem 3.5 are holomorphic, and furthermore (using the ampleness

of ω), the constant k can be chosen large enough that V2 can be embedded as a sub-

bundle of Vc2Δ1 . The quotient bundle can be identified with the difference bundle

E , which therefore has a holomorphic structure depending on the above embedding;we now show that it is possible to arrange that E , endowed with this structure, ispolystable. (By stability we shall mean μ-stability w.r.to the polarisation ω.)

Let us start by recalling that V1 = L1 ⊗Lkω ⊕L−11 ⊗Lkω, and V2 = L2 ⊕L−1

2 .Choose a large enough integer k1 such that Lk1ω that is very ample, and let Cbe a general curve cut out by three sections of this line bundle. It follows fromProposition 3.4 that d ≡ degree L−1

2 ⊗ L1 ⊗ Lkω|C = degree L2 ⊗ L−11 ⊗ Lkω|C =

degree Lkω|C = kk31 < ω4 >, and will assume that k is chosen such that d >2genus(C) = 3k41 < ω4 > +2. We next make the following assumption:

(7.1) dim H0(C,Lkω|C) = c2Δ

which we will return to below. Let W denote a subspace of H0(X ′, L−12 ⊗L1⊗Lkω),

chosen such that

• the restriction map W → H0(C,L−12 ⊗ L1 ⊗ Lkω|C) is an isomorphism,

and• W is base-point free.

Consider now the evaluation map E : W ⊗OX′ → L−12 ⊗ L1 ⊗ Lkω, and let F be

the kernel; by construction F fits in the exact sequence

0 → F → W ⊗OX′ → L−12 ⊗ L1 ⊗ Lkω → 0 .

By Butler’s Theorem ([Bu]), the restriction of F to C is stable, and this proves thatF itself is stable. We next choose a subspace U of H0(X ′, L2 ⊗ L−1

1 ⊗ Lkω) withsimilar properties and obtain a second stable bundle G that fits in the sequence

0 → G → U ⊗OX′ → L2 ⊗ L−11 ⊗ Lkω → 0

Dualising, tensoring by suitable line bundles and adding the two sequences, we get

0 → V2 → Vc2Δ1 → F ⊗ L1 ⊗ Lkω ⊕ G ⊗ L−1

1 ⊗ Lkω → 0

where F denotes the dual of F and G denotes the dual of G, and we have used theassumption (7.1), namely, dim W = dim U = c2Δ. Repeatedly using Proposition3.4 we see that the two summands in the last sum have the same slope. Considernow the assumption (7.1). By Riemann-Roch, this is equivalent to:

(kk31 − (3/2)k41) < ω4 >= c2Δ

This is solved by taking

k = (c2Δ

k31+

3k12

)/ < ω4 >

169

16 T. R. RAMADAS

This is where the choice of c comes in - we choose c and k1 such that k is an integer(and large enough). Once this is done

Theorem 7.1. The bundle E (on X ′) can be given a holomorphic structuresuch that it is polystable.

The above application of Butler’s theorem is inspired by its use in [M].By Donaldson-Uhlenbeck-Yau, such a bundle would admit a Hermite-Einstein

metric and therefore a Spin(7) instanton.

References

[A-H] M.F. Atiyah and F. Hirzebruch, Vector bundles and homogeneous spaces, Prec. Sympos.Pure Math. 3 (1961), 7–38.

[A-O] B.S. Acharya and M. O’Loughlin, Selfduality in d=8 dimensional euclidean gravity, Phys.

Rev. D 55 (1997), 4521–4524.[B-K-S] L. Baulieu, H. Kanno and I.M. Singer, Special quantum field theories in eight and other

dimensions, Commun. Math. Phys. 194 (1998), 149–175.[B6] K. Becker, M. Becker, D.R. Morrison, H. Ooguri, Y. Oz, Y. Zheng, Supersymmetric cycles

in exceptional holonomy manifolds and Calabi-Yau four-folds, Nuclear Physics B, 480(1996), 225-238.

[B] S. Brendle, Complex anti-self-dual instantons and Cayley submanifolds,math.DG/0302094, 2003.

[Bu] D.C. Butler, Normal generation of vector bundles over a curve, J. Differential Geom. 39(1994), 1–34.

[D-T] S. K. Donaldson and R. P. Thomas, Gauge theory in higher dimensions, in “The Geomet-ric Universe; Science, Geometry, And The Work Of Roger Penrose”, Oxford UniversityPress, (1998).

[H-L] F. R. Harvey and H. B. Lawson, Jr, Calibrated geometries. Acta Mathematica, 148(1982), 47–157.

[J] D.C. Joyce, Riemannian Holonomy Groups and Calibrated Geometry, Oxford GraduateTexts in Mathematics (2007).

[L] C. Lewis, Spin(7) Instantons, Oxford University D.Phil. thesis (1998).[P] H. Pohlmann, Algebraic cycles on abelian varieties of complex multiplication type, Ann.

of Math. 88 (1968), 161–180.[M] E. C. Mistretta, Stable vector bundles as generators of the Chow ring, Geom. Dedicata

117 (2006), 203–213.[S] A. Sommese, Quaternionic manifolds, Math. Ann. 212 (1975), 191–214.

[T] G. Tian, Gauge theory and calibrated geometry I, Ann. of Math. 151 (2000), 193–268.[V] C. Voisin, Some aspects of the Hodge conjecture, Japanese Journal of Mathematics 2 2

(2007), 261–296.

Abdus Salam I.C.T.P., 11 Strada Costiera, Trieste 34014, Italy


170


Remarks on Parabolic Structures

Seshadri

Abstract. Let Y be a smooth projective curve with an action of a finitegroup π and X = Y mod π. Let G be a reductive group and P −→ Y a(π − G) bundle i.e. P is a principal G-bundle on Y such that the action ofπ on Y lifts to an action on P commuting with the canonical action of G onP . The question is to give an equivalent description of (π − G) bundles on

Y by suitable objects on X. When G is the full linear group, the answer isgiven by principal G-bundles on X, endowed with what are called parabolicstructures at the ramification points of X. This notion of parabolic structuresgeneralizes in an obvious manner, to the case of principal G-bundles on Xwhen G is a general reductive. However, in the case of a general reductivegroup, the required description by objects turns out to be subtler than thisobvious generalization and gets connected with Bruhat-Tits theory.

1. Introduction

Let Y be a smooth projective curve (defined over C) with an (effective) actionof a finite group π, X the smooth projective curve X = Y mod π and p : Y −→ Xthe canonical quotient morphism. Let G be a reductive algebraic group and P a(π−G) bundle on Y i.e. P is a principal G-bundle over Y and the action of π on Ylifts to an action on P commuting with the (right) action of G on P . The questionis to give an equivalent description of the moduli problem of (π−G) bundles on Yby objects on X. Recall that if G is the full linear group, this required descriptionis given by principal G-bundles on X having parabolic structures at the ramificationpoints of X i.e. the points of X over which p is ramified. In fact P is equivalentlydescribed by a π-vector bundle V on X and if W = pπ∗ (V ) is the invariant directimage of V , the parabolic structure is described by giving flags of fixed type on thefibres of W at the ramification points of X ([6]). In the general case the descriptionby objects on X (given in Remark 2.13 below) turns out to be somewhat subtlerand it gets intimately connected with Bruhat-Tits theory (see [3]). For example,it does not seem to be possible, in general, to associate naturally to P a principalG-bundle on X (see Case II, §3 below; also the connection with vector bundles withorthogonal and symplectic structures having singularties at the ramification pointsin X); however, the moduli stack of these objects seems to be related to the modulistack of principal G-bundles by a “Hecke correspondence” (see Remark 2.13 below

2010 Mathematics Subject Classification. 14D20, 14H60.


1



171

C. S.

2 C.S. SESHADRI

and [7]). In this talk we shall illustrate this phenomenon for the case of classicalgroups. These objects appear in the study of compactifications of moduli spaces ofprincipal G-bundles on nodal curves (see [9]). They are also related to the work[8], which we came to know recently. See also the more recent preprint [5].

These remarks have taken shape in the course of numerous discussions with V.Balaji and are inspired by the classical work of A. Weil [11]. In a subsequent articlea description of the above mentioned objects on X in terms of Bruhat-Tits theoryand a correspondence between these objects and (π −G)-bundles, will be given.

Recall that the moduli spaces of (π −G)-bundles on Y have been constructedby Tannakian methods (see [1], [2]).

2. (π −G) bundles on Y and lattices on X

Let x ∈ X be a ramification point of p : Y −→ X. Let Ux denote the groupof (π−G) automorphisms of the restriction of P to a sufficiently small π-invariantneighbourhood of p−1(x). We call Ux the unit group at x. The main point is todescribe this as an object on X.

Let y be a point of Y lying over a ramification point x ∈ X. Let Γ be theisotropy subgroup of π at y. We can find sufficiently small neighbourhoods E andD of y and x respectively such that E is Γ-invariant and D = E mod Γ. Wecan suppose that E and D are discs with centres (0) which correspond to y and xrespectively.

Let τ be a coordinate function on E so that z = τ r is a coordinate function onD, where r is the order of the cyclic group Γ. The restriction of P to E is a (Γ−G)bundle and one sees that Ux is just the group of (Γ−G) automorphisms of P overE, which identifies with the trivial bundle E×G (G action given by multiplicationon the right).

Recall the fact that the Γ-action on E×G is given by a representation ρ : Γ −→G (see [11], [4], [10]) i.e.

γ · (u, g) = (γu, ρ(γ)g), u ∈ E, γ ∈ Γ.(2.1)

Let φ0 ∈ Ux i.e.

φ0 : E ×G −→ E ×G is a (Γ−G) automorphism.(2.2)

We see thatφ0(u, g) = (u, φ(u)g)

where φ : E −→ G is a regular map satisfying the following:

φ(γ · u) = ρ(γ)φ(u)ρ(γ)−1, u ∈ E.(2.3)

We see then that the functions φ as above identify with the elements of Ux.Let us fix a maximal torus T of G. Now T is a product of one dimensional tori.

We represent it in the diagonal form as follows (or if we wish we could embed G ina full linear group and work with matrices)

T =

⎡⎢⎢⎣

t1 0.

.0 tm

⎤⎥⎥⎦ .(2.4)

C. S. SESHADRI172

PARABOLIC BUNDLES 3

Then we can suppose that the representation ρ of Γ in G factors through T (by asuitable conjugation). Fix a generator γ in Γ. We can suppose that the coordinatefunction τ of E is so chosen that γ · τ = ζτ , where ζ is a (primitive) rth of unity.Then ρ takes the form

ρ(γ) =

⎡⎢⎢⎣

ζa1 0.

.0 ζam

⎤⎥⎥⎦ with ai ∈ Z.(2.5)

We can suppose that |ai| < r for all i (or even 0 ≤ ai < r) and take

αi = ai/r, so that |αi| < 1.(2.6)

We denote the function τai by zαi (z = τ r). Define the “meromorphic” (or“rational”) map Δ : E −→ T (morphism on the punctured disc E \ (0)):

Δ =

⎡⎢⎢⎣

τa1 0.

.0 τam

⎤⎥⎥⎦ =

⎡⎢⎢⎣

zα1 0.

.0 zαm

⎤⎥⎥⎦(2.7)

Then we have

Δ(γu) = ρ(γ)Δ(u), u ∈ E(2.8)

where Δ can be taken as a function Δ : E −→ G (through T ↪→ G).Now define the function ψ : E −→ G by

ψ = Δ−1φΔ(2.9)

That is,

φ = ΔψΔ−1.

Then we see that ψ(γu) = ψ(u) so that ψ goes down to a function D −→ Gand we denote this by the same letter ψ. We see that ψ is a meromorphic map.We see then that the unit group Ux can be identified with the set of meromorphicmaps ψ : D −→ G defined above.

It is customary to work with complete local rings. Let Oy (resp. Ox) denote thecompletion of the local ring of Y at y (resp. of X at x). Then φ defines an elementof G(Oy) (the group of Oy valued points of G). We write simply φ ∈ G(Oy). Notethat the action of Γ on E × G induces a canonical action of Γ on G(Oy) (can bededuced from (2.3)) and then the unit group Ux is just G(Oy)

Γ. Let Kx (resp.Ky) be the quotient field of Ox (resp. Oy). We see that ψ ∈ G(Kx). Then by themap φ �−→ ψ, we can identify Ux with a subgroup of G(Kx); however, the choiceof Δ satisfying (2.8) is not unique and if we choose another, we see easily that theresulting one is a conjugate subgroup in G(Kx). Thus we have (see Remark 2.12below):

Ux determines a lattice or bounded subgroup in G(Kx), well-determined upto conjugacy. We denote this by the same letterUx.

(2.10)

Let us now take G = GL(m). Then we can write φ = ||φij(τ )||, ψ =||ψij(z)||, 1 ≤ i, j ≤ m (as matrices). Then (2.9) takes the form

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4 C.S. SESHADRI

φij(τ ) = ψij(z)zαi−αj .(2.11)

We can suppose that 0 ≤ α1 ≤ α2 ≤ · · · ≤ αm < 1. Since |αi − αj | < 1, we deduceeasily that ψij are regular i.e. Ux ⊂ G(Ox). To see this suppose that ψij is notregular. Then considered as a function in τ (z = τ r), ψij has a pole of order ≥ r,whereas zαi−αj could have only a zero of order strictly less than r (as a function inτ ). But φij(τ ) is regular, which leads to a contradiction.

Besides, we see also that ψij(0) is a lower triangular matrix. This fact gives theparabolic (or rather quasi-parabolic) structure on the vector bundle pπ∗ (V ) = W ,V being the π-vector bundle associated to the principal bundle P (for details see(§1 of [6]), also ([11] esp. p.194).

Remark 2.12. Let L be a p-adic local field and A the ring of integers inL (A is a complete local ring and L is its quotient field). Then G(L) is locallycompact (we have to make suitable hypothesis on the field of definition for G).Then G(A) is a maximal compact subgroup of G(L) and unlike the real case, allthe maximal compact subgroups need not be conjugate. Compact subgroups ofG(L) are often referred to as lattices and the maximal ones as maximal lattices.In the geometric case that we are considering, G(Kx) is not locally compact andthe notion of bounded subgroups generalises that of compact subgroups (followingBruhat-Tits [3]). A subgroup of H of G(Kx) is said to be bounded if the “order ofpole” of an element of H is bounded by a universal constant (taking an embeddingof H in a full-linear group so that an element of H is represented by a matrix withentries in Kx, this notion makes sense). We observe that Ux is a bounded subgroupof G(Kx) in this sense. This is an easy consequence of (2.9) noting that φ is regular.Of course G(Ox) is a bounded subgroup; in fact it is a maximal bounded subgroupof G(Kx).

Let j : G(Ox) −→ G be the “evaluation map” i.e. the map G(Ox) −→ G(C)induced by the canonical mapOx −→ residue field which is C in our case. We denoteG(C) by G. The inverse image by j of a Borel subgroup in G is an example of an“Iwahori subgroup” in G(Kx). More generally an Iwahori subgroup of G(Kx) is, bydefinition, a conjugate of this particular Iwahori subgroup. A parahoric subgroupof G(Kx) is, by definition, a subgroup which contains an Iwahori subgroup. Fixan Iwahori subgroup I. Then one knows that there are only a finite number rof maximal bounded subgroups of G(Kx) containing I and any maximal boundedsubgroup is a conjugate of precisely one containing I. If G is the full linear group,then one knows that r = 1, so that a maximal bounded subgroup of G(Kx) isalways a conjugate of G(Ox). This also explains why Ux is contained in G(Ox) inthis case.

Remark 2.13. We shall now describe (π−G) bundles on Y as suitable objectson X.

For simplicity we suppose that x is the only ramification point and that G issemi-simple.

The description of (π−G) bundles on Y by objects onX is given in (2.26) belowand generalises a well-known description of principal G-bundles as a double cosetspace (or an equivalent adelic description). The point of this well-known descriptionis that the restriction of a principal G-bundle to D, as well as X \ {x}, is trivialand hence it is described by a single transition function i.e. a map D \ (0) −→ G

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PARABOLIC BUNDLES 5

such that isomorphism classes of G-bundles are interpreted as equivalence classes oftransition functions. This argument generalises in a fairly easy manner to the caseof (π−G) bundles on Y with, for example, a transition function being interpretedas a Γ-equivariant map E \ (0) −→ G. The details are given below.

Two (π − G) bundles on Y are said to be locally isomorphic at x if they areisomorphic as (π−G) bundles over p−1(D) = V1, D a sufficiently small disc neigh-bourhood of x as above. We see that two such bundles are locally isomorphic atx if and only if their restrictions to E are isomorphic as Γ-bundles. Recall that Eis a Γ-invariant neighbourhood of y, y being a point of Y lying over x. We cansuppose that V1 is a disjoint union of discs of the form E i.e. each disc contains aunique point of Y lying over x. Hence if P is a (π − G) bundle on Y , we call therestriction of P to E, which is a (Γ −G) bundle, as the local invariant of P at x.We shall now fix this local invariant i.e. we fix y and suppose that the restrictionP to E is the (Γ−G) bundle with the action of Γ on E ×G given as in (2.1) i.e.

P |E � E ×G with the action of Γ given byγ · (u, g) = (γu, ρ(γ)g), u ∈ E and g ∈ G

(2.14)

ρ : Γ −→ G being a representation. We can also say that the local invariant isgiven by the representation ρ (fixing y) of the isotropy subgroup of π at y. Wedenote M(ρ) by the moduli space (or rather moduli stack) of isomorphism classesof (π −G) bundles with local invariant ρ.

Let X1 denote the complement of x in X and Y1 = p−1(X1). Now π acts freelyon Y1 so that the restriction of P to Y1 goes down to a principal G-bundle on X1

which is trivial (in the algebraic sense) since G is semi-simple. Hence we have:

P |Y1� Y1 × G with the action of π given by γ · (u, g) =

(γu, g), γ ∈ π and u ∈ Y1.(2.15)

Let us call the restriction of P to V1 (resp. Y1) by P1 (resp. P2). We note that therestriction of P1 to E is just the one described in (2.14) and P2 is the one describedin (2.15).

The (π − G) bundle P is given by a “transition function” i.e. a (π − G)isomorphism (holomorphic):

θ : P2|V1∩Y1−→ P1|V1∩Y1

.(2.16)

Now if s is a π-invariant (i.e. π-equivariant) rational section of P over Y (whichexists, since for example; the restriction of P to Y1 descends to the trivial G-bundleon X1 in the algebraic sense). Then if s1 (resp. s2) is the restriction of s to asection of P1 over V1 (resp. a section of P2 over Y1), note that by the isomorphismθ in (2.16) above, s2 is taken to s1. Besides by (2.15) above, s2 defines a morphismY1 −→ Y1×G given by u −→ (u, s∗2(u)) so that s∗2 defines a rational (in the algebro-geometric sense) map Y −→ G which is regular on Y1. Of course s∗2 is π-invarianti.e. s∗2(γ · u) = s∗2(u), γ in Γ. Thus we have

(i) θ(s2) = s1 over V1 ∩ Y1

(ii) s2 defines (through the identification (2.15)) a π-invariantrational map s∗2 : Y −→ G which is regular on Y1.

(2.17)

Observe that if Q is any (π−G) bundle in M(ρ), then Q|V1� P1 and Q|Y1

� P2

as (π −G) bundles by (2.14) and (2.15) above (noting that Q|V1is determined by

175

6 C.S. SESHADRI

Q|E). Thus Q is defined by an isomorphism as in (2.16) above. Let us denote it byφ. Then P is (π −G) isomorphic to Q if and only if we have the following:

λ θ μ = φλ a (π −G) automorphism of P1

μ a (π −G) automorphism of P2.(2.18)

Observe that μ is given by a morphism (using (2.15)):

Y1 ×G −→ Y1 ×G,(u, g) −→ (u, μ∗(u)g),μ∗(γ · u) = μ(u), γ ∈ π i.e.μ∗ goes down to morphism X1 −→ G.

(2.19)

Note that the punctured disc E∗ = E \ (0) is contained in V1 ∩ Y1 and the (π−G)isomorphism θ is completely characterized by its restriction to E∗. We observe by(2.15) that the restriction of P2 to E∗ is the (Γ−G) bundle E∗ ×G over E∗ withthe action of Γ given by

γ : E∗ ×G −→ E∗ ×G, γ ∈ Γγ(u, g) = (γu, g).

(2.20)

The restriction of P1 to E∗ is the (Γ−G) bundle E∗ ×G on E∗ with the action ofΓ given by

γ : E∗ ×G −→ E∗ ×Gγ(u, g) = (γu, ρ(γ)g), γ ∈ Γ.

(2.21)

Then the restriction of θ to E∗ is then a (Γ − G) isomorphism of the bundle in(2.20) with the one of (2.21). We see easily that θ is defined by:

E∗ ×G −→ E∗ ×G(u, g) −→ (u, θ∗(u)g)θ∗(γ · u) = ρ(γ)θ∗(u)

θ∗ : E∗ −→ G.

(2.22)

Observe that the map Δ, as in (2.8), has the property Δ(γ · u) = ρ(γ)Δ(x). Thuswe can write

θ∗ = Δθ∗∗ such thatθ∗∗(γu) = θ∗∗(u) i.e. θ∗∗ goesdown to a regular map D∗ −→ G, D∗ = D \ (0).

(2.23)

We claim that

θ∗∗ extends to a meromorphic map E −→ G and hence by 2.23descends to a meromorphic map D −→ G which is regular onD∗.

(2.24)

This is an easy consequence of (2.17) as follows. The restriction of s1 to E definesa regular map E −→ E × G, u �−→ (u, s∗1(u)), where s∗1 : E −→ G is a regularmap. The restriction of s∗2 to E∗ is regular and extends to a meromorphic mapE −→ G by the property (ii) of (2.17). Further (i) of (2.17) translates to therelation θ∗ = s∗1 · (s∗2)−1, so that θ∗ is meromorphic in E and hence θ∗∗ is alsomeromorphic in E, which proves (2.24).

Now the (π−G) automorphisms of P1 identify with the (Γ−G) automorphismsof the restriction of P1 to E i.e. the (Γ − G) bundle on E as in (2.14). Thus λidentifies with an element λ∗ of the unit group at x (i.e. an element as in (2.2)

C. S. SESHADRI176

PARABOLIC BUNDLES 7

above, before its identification as a subgroup of G(Kx)). Thus the equivalencerelation (2.18) translates into the following form:

λ∗(Δθ∗∗)μ∗ = Δφ∗∗ i.e.(Δ−1λ∗Δ)θ∗∗μ∗∗ = φ∗∗.

(2.25)

Let us denote by G(Khx ) the set of germs of regular maps D∗ −→ G which

extend to meromorphic maps D −→ G. We have an inclusion G(Khx ) ⊂ G(Kx).

Set Uhx = Ux ∩ G(Kh

x ) i.e. we work in the holomorphic and meromorphic set upinstead of the “formal” set up. Observe that θ∗∗ ∈ G(Kh

x ) and (Δ−1λ∗Δ) ∈ Uhx .

Further μ∗ is a regular map X1 −→ G extending to a meromorphic map X −→ Gso that μ∗ is a rational map i.e. μ∗ ∈ G(X1). Thus from (2.25) we deduce thefollowing:

The moduli stack M(ρ) identifies with the double coset space

Uhx \G(Kh

x )/G(X1).(2.26)

In fact, it can be shown that the moduli stack M(ρ) can be identified with

(i) Ux\G(Kx)/G(X1) or in adelic language as follows:

(ii)∏

p∈X Cp\G(A)/G(K)

where A denotes group of the adeles, K the function field of X, Cp = Op for p = xand Cx = Ux.

The moduli stack M(ρ) depends only on the unit group Ux and not on theexplicit nature of the representation ρ. We can then denote M(ρ) as M(Ux). Themoduli stack M(Ux) is to be understood as the moduli stack with “quasi-parabolicstructures” in the sense of [6], where the representation ρ gives weights to defineparabolic structures and define stability and semistability, leading to the construc-tion of moduli spaces of semistable parabolic vector bundles as projective varieties.

Note that the moduli stack M(Ux) makes sense, say for any bounded subgroupUx of G(Kx). It is likely that there are quasi-projective schemes associated to thesemoduli stacks (under suitable hypotheses). These could be called moduli stackswith level structures.

Suppose that we have an inclusion U ′x ⊂ Ux of bounded subgroups of G(Kx).

Then we have a canonical morphism of stacks

M(U ′x) −→ M(Ux).

Note that if Ux = G(Ox), then the moduli stack M(Ux) is just the moduli stackM of principal G-bundles on X. If we have Ux ⊂ G(Ox) then we have a canonicalmorphism of stacks

M(Ux) −→ M

i.e. we have an underlying principal G-bundle to a (π − G) bundle in the caseUx ⊂ G(Ox). As we saw above this happens in the case of the full linear group.

Suppose that Ux is a parahoric subgroup in G(Kx). Then by a suitable conju-gation we can suppose that we have inclusions

I ⊂ Ux and I ⊂ G(Ox),

I being an Iwahori subgroup. This leads to the following diagram of morphisms ofstacks:

M(I)↙ ↘

M M(Ux)

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8 C.S. SESHADRI

i.e. M and M(Ux) are related by a “Hecke correspondence” (see [7]). We shall seeexamples of these in §3 below.

It seems likely that the moduli stacks M(Ux) coming from (π−G) bundles (asabove) are related to M by “Hecke correspondences which are good” in the sensethat there are quasi-projective schemes associated to these correspondences.

3. The case of G = SO(2n) or Sp(2n)

Let us now consider the case when G = SO(2n) (resp. Sp(2n)). Define

J∗ =

[0 JJ 0

](respectively

=

[0 J−J 0

])

where J is the (n× n)–matrix with 1’s on the antidiagonal and zero elsewhere:

J =

⎡⎢⎢⎢⎢⎢⎢⎣

0 1.

1.

.1 0

⎤⎥⎥⎥⎥⎥⎥⎦n×n

.

We take SO(2n) (resp. Sp(2n)) as the subgroup of GL(2n) which leaves J∗

invariant i.e. formed by S in GL(2n) such that SJ∗ tS = J∗. Then a maximaltorus T of G can be taken in the diagonal form

T =

⎡⎢⎢⎢⎢⎢⎢⎣

t1 0.

tnt−1n

.0 t−1

1

⎤⎥⎥⎥⎥⎥⎥⎦.(3.1)

One knows that G ∩ B+ (resp. G ∩ B−) is again a Borel subgroup in G, B+ =(resp. B−) = upper (resp. lower) triangular matrices in GL(2n).

Let us consider the case where the ai and the associated αi in (2.6) can betaken in the following form:

(3.2) −1 < −α1 ≤ −α2 ≤ · · · ≤ −αn ≤ αn ≤ αn−1 ≤ · · · ≤ α1 < 1,

with (0 ≤ αi < 1)We write β1 = −α1, β2 = −α2, · · ·βn = −αn, βn+1 = αn, · · · , β2n = α1.Thus we have in this case diagonal matrices:

A =

⎡⎢⎢⎣

z−α1 0.

.0 z−αn

⎤⎥⎥⎦ =

⎡⎢⎢⎣

zβ1 0.

.0 zβn

⎤⎥⎥⎦ ,(3.3)

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PARABOLIC BUNDLES 9

B =

⎡⎢⎢⎣

zαn 0.

.0 zα1

⎤⎥⎥⎦ =

⎡⎢⎢⎣

zβn+1 0.

.0 zβ2n

⎤⎥⎥⎦ .(3.4)

Let

Δ =

[A 00 B

]

Thus we get

φ = ΔψΔ−1 i.e.φij(τ ) = ψij(z)z

βi−βj(3.5)

and φ, ψ and Δ leave J∗ invariant. Let I denote the subgroup of G(Ox) such thatby the evaluation map G(Ox) −→ G, I is the inverse image of the Borel subgroupinduced by the lower triangular matrices in GL(2n). Then I is an Iwahori subgroupin G(Kx). We note that |βi − βj | < 2 and ψij can have at most a pole of order 1.We need to consider three cases:

Case I: |βi − βj | < 1

Case II: |βi − βj | ≤ 1 and the value 1 is attained

Case III: |βi − βj | > 1 for some i, j

In Case I, by the same argument as for the full linear group (βi’s are increas-ing), we deduce that ψij(z) are regular and ψij(0) is lower triangular. (βi’s areincreasing).

In Case II, the ψij(z) can have poles of order 1 and in Case III, the order ofthe pole is at most 1 but they can have zeroes of order 2.

Thus we get, in Case I,G(Ox) ⊃ Ux ⊃ I

In Case II, we have

G(Kx) ⊃ Ux ⊃ I but Ux is not contained in G(Ox).

We observe that, in the cases I and II, Ux is a parahoric subgroup of G(Kx), butin case II, Ux is not even conjugate (in G(Kx)) to a subgroup of G(Ox), in particularthe maximal lattice containing Ux is not conjugate to the maximal lattice G(Ox).

In Case III, in contrast, we have of course G(Kx) ⊃ Ux but Ux does not containI. It is not clear whether Ux is a parahoric subgroup of G(Kx).

Let M,M(I) and M(Ux) be the moduli stacks of (π − G) bundles on Y withlocal invariants defined by the unit group G(Ox), I and Ux respectively (see towardsthe end of Remark 2.13).

Now we have the following:Case I: |βi − βj | < 1

We have a morphism of stacks:

M(I) −→ M(Ux) −→ M(3.6)

(where M = M(G(Ox)) - set of isomorphism classes of G-bundles on X).Note that this is similar to the case of the full linear group. An object in M(Ux)

is to be viewed as having a parabolic (strictly speaking a quasi-parabolic) structureover the image of this object in M , which is again a G-bundle on X.

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10 C.S. SESHADRI

Case II: |βi − βj | ≤ 1 and the value 1 is attained

We have morphisms of stacks

M(I)↙ ↘

M M(Ux)(3.7)

induced by the inclusions I ⊂ G(Ox) and I ⊂ Ux.The above diagram means that M and M(Ux) are related by a “Hecke corre-

spondence” (see [7]). There seems to be no natural way of associating a G-bundleon X to an object in M(Ux) in this case II.

Let αi be as in (3.2) above and let us define δi, 0 ≤ δi < 1, 1 ≤ i ≤ 2n, asfollows:

(3.8){δi = 1− αi if αi = 0 and δi = 0 if αi = 0, 1 ≤ i ≤ n,

δ2n+1−j = αj , 1 ≤ j ≤ n}.Set

A1 =

⎡⎢⎢⎣

zδ1 0.

.0 zδn

⎤⎥⎥⎦(3.9)

B1 =

⎡⎢⎢⎣

zδn+1 0.

.0 zδ2n

⎤⎥⎥⎦(3.10)

Now let

Δ1 =

[A1 00 B1

]

Define θ as follows:

φ = Δ1θΔ−11 ⇐⇒ φij(z) = θij(z)z

δi−δj .(3.11)

We see that

θ = L · ψ · L−1 with L = Δ−11 ·Δ

and θ leaves invariant the form Q = LJ∗L i.e

θ Q tθ = Q

observing that Lt = L.Since |δi−δj | < 1 we find that θij(z) are regular functions i.e. θ ∈ GL(2n)(Ox)

(as in (2.11) above, the order of the δi’s do not play a role at this point).Let k = #{i|αi = 0}. We find that

L =

[z−1Ik×k 0

0 I2n−k×2n−k

]

We see that Q is a quadratic (resp. symplectic) form over Kx with a singularityat x. Let Uk

x be the unit group ofQ i.e. consisting of all the elements ofGL(2n)(Ox),which leave Q invariant. We see that, by conjugation in GL(2n)(Kx), Ux can beidentified with a subgroup of Uk

x and coincides with Ukx for the choice αi = 1/2,

C. S. SESHADRI180


1 ≤ i ≤ k. The groups Ukx seem to be maximal parahoric and mutually non-

conjugate.The (π−G) bundle P can be equivalently described by a rank 2n vector bundle

V on X, endowed with a π-invariant, non-degenerate quadratic (resp. symplectic)form. Then if W = pπ∗ (V ), W acquires a quadratic (resp. symplectic) form whichis non-degenerate outside x and described by Q locally at x. Thus the moduli stackof (π −G) bundles is described equivalently as the moduli stack of vector bundlesW on X with a non-degenerate quadratic (resp. symplectic) form outside x andwith singularity at x of type given by Q as above.

4. The case when G = SL(n)

Let us now take G = SL(n). Fix k, 1 ≤ k ≤ n − 1. Then we can take thechoice of αi or δi (as in (3.2) and (3.8) above) as follows:

α1 = · · · = αk = − (n− k)

n(4.1)

αk+1 = · · · = αn = k/n(4.2)

or equivalentlyδ1 = · · · = δn = k/n.

We denote the corresponding unit group by Ukx . For k = 0, we take U0

x =SL(n)(Ox). These groups are maximal parahoric.

Then we see that Ukx ⊂ SL(n)(Kx) (1 ≤ k ≤ n− 1), and takes the form

Ux =

[A zB

z−1C D

]

where the entries of A,B,C,D are in Ox, and A is a (k × k)-matrix, and D an(n− k)× (n− k)-matrix.

We see that the lattices Ukx are not conjugate in SL(n)(Kx). They are maximal

parahorics. The moduli space M(Ukx ) is equivalently described as the moduli space

of vector bundles on X of rank n whose determinant is a fixed line bundle of degreek (in particular for 1 ≤ k ≤ n− 1, they are not SL(n) bundles on X) and they areall related by Hecke correspondences (see [7]). For general {αi}, the moduli spacecan then be viewed as a moduli space with quasi-parabolic structures in the usualsense on a vector bundle on X.

References

[1] V. Balaji, I. Biswas and D.S. Nagaraj, Principal bundles over projective manifolds withparabolic structure over a divisor. Tohuku Math. Jour. 53 (2001), 337–368.

[2] V. Balaji, I. Biswas and D.S. Nagaraj, Ramified G-bundles as parabolic bundles, Jour. Ra-manujan Math Society, 18, No 2 (2003) 123-138.

[3] F. Bruhat and J. Tits, Groupes reductifs sur un corps local II, Schemas en groupes, Existence

d’une donnee radicielle valuee, Inst. Hautes Eludes Sci. Publ. Math. 60 (1984), p.197–376.[4] A. Grothendieck, Sur la memoire de Weil “Generalisation des fonctions abeliennes”,

Seminaire Bourbaki, Expose 141, (1956-57).[5] J. Heinloth, Uniformization of G–bundles, arXiv: 0711.4450.[6] V. Mehta and C.S. Seshadri, Moduli of vector bundles on curves with parabolic structures,

Math. Ann 248 (1980), 205–239.[7] M.S. Narasimhan and S. Ramanan - The papers [13], [18] and [19] in the Collected papers of

M.S. Narasimhan - Hindustan Book Agency.[8] G. Pappas and M. Rapoport, Some questions about G-bundles on curves, Mathematisches

Institut der Universitat Bonn, November 2008.

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12 C.S. SESHADRI

[9] C.S. Seshadri. Moduli spaces of torsion free sheaves on nodal curves and generalisations - I,to appear in the volume dedicated to P.E. Newstead.

[10] C. Teleman and C.Woodward, Parabolic bundles, products of conjugacy classes, and quantumcohomology, Annales Institut Fourier, Vol. 53, (2003), no. 3, p. 713-748.

[11] A. Weil - Generalisations des fonctions abeliennes - Collected papers, Vol. I, Springer-Verlag.

Chennai Mathematical Institute, SIPCOT IT Park, Padur Post, Siruseri 603103,

Tamil Nadu, India


C. S. SESHADRI182


Iterated Destabilizing Modifications for Vector Bundles withConnection

Carlos Simpson

Abstract. Given a vector bundle with integrable connection (V,∇) on acurve, if V is not itself semistable as a vector bundle then we can iterate aconstruction involving modification by the destabilizing subobject to obtain aHodge-like filtration F p which satisfies Griffiths transversality. The associatedgraded Higgs bundle is the limit of (V, t∇) under the de Rham to Dolbeaultdegeneration. We get a stratification of the moduli space of connections, withas minimal stratum the space of opers. The strata have fibrations whose fibersare Lagrangian subspaces of the moduli space.

1. Introduction

Suppose X is a smooth projective curve over C. Starting with a rank r vectorbundle with integrable holomorphic connection (V,∇), if V is semistable as a vectorbundle, we get a point in the moduli space U(X, r) of semistable vector bundles ofrank r and degree 0 on X.

Let MDR(X, r) denote the moduli space of vector bundles with integrable con-nection of rank r. The open subset G0 where the underlying vector bundle is itselfsemistable thus has a fibration

MDR(X, r) ⊃ G0 → U(X, r).

The fiber over a point [V ] ∈ U(X, r) (say a stable bundle) is the space of connec-tions on V , hence it is a principal homogeneous space on H0(End(V ) ⊗ Ω1

X) ∼=H1(End(V ))∗ = T ∗

V U(X, r). So, the above fibration is a twisted form of the cotan-gent bundle T ∗

V U(X, r) → U(X, r). At points where the bundle V is not semistable,we will extend G0 to a stratification of MDR(X, r) by locally closed subsets Gα.

If V is not semistable, let H ⊂ V be the maximal destabilizing subsheaf. Recallthat H is a subsheaf whose slope μ(H) = deg(H)/rk(H) is maximal, and amongsuch subsheaves H has maximal rank. It is unique, and is a strict subbundle so thequotient V/H is also a bundle.

2010 Mathematics Subject Classification. Primary 14H60; Secondary 14D07, 32G34.Key words and phrases. Connection, Deformation, Higgs bundle, Moduli space, Oper, Re-

ductive group.This research is partially supported by ANR grants BLAN08-1-309225 (SEDIGA) and

BLAN08-3-352054 (G-FIB).


1



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2 C. SIMPSON

The connection induces an algebraic map

θ = ∇ : H → (V/H)⊗ Ω1X .

Define a Higgs bundle (E, θ) by setting E1 := H,E0 := V/H, E := E0⊕E1, and θis the above map. It is a “system of Hodge bundles”, that is a Higgs bundle fixedby the C

∗ action.If (E, θ) is a semistable Higgs bundle, the process stops. If not, we can continue

by again choosing H ⊂ (E, θ) the maximal destabilizing sub-Higgs-bundle, thenusing H to further modify the filtration according to the formula (3.2) below. Atthe end of §3 the proof of our first main Theorem 2.5 consists in showing that thisrecursive process stops at a Griffiths-transverse filtration of (V,∇) such that theassociated graded Higgs bundle is semistable.

Classically filtered objects (V,∇, F •) arose from variations of Hodge structure.In case an irreducible connection supports a VHS, our iterative procedure constructsthe Hodge filtration F • starting from (V,∇).

In general the filtration F • given by Theorem 2.5 is not unique, see Proposition4.3. The associated-graded Higgs bundle (E = Gr•F (V ), θ) is unique up to S-equivalence, as follows from the following limit point interpretation.

In terms of the nonabelian Hodge filtration [54] MHod → A1 the above processgives a way, described in §4, of calculating the limit point limt→0(tλ, V, t∇). Thelimit is a point in one of the connected components of the fixed point set of the Gm

action on the moduli space of Higgs bundles MH. Looking at where the limit landsgives the stratification by Gα ⊂ MDR. Existence of the limit is a generalizationto MDR of properness of the Hitchin map for MH. The interpretation in terms ofGriffiths-transverse filtrations was pointed out briefly in [54].

Given the S-equivalence class, it makes sense to say whether (E, θ) is stable ornot. Proposition 4.3 shows that the filtration F • is unique up to shifting indices, ifand only if (E, θ) is stable. In §6, the nonuniqueness of the filtration is related toa wall-crossing phenomenon in the parabolic case.

In the present paper, after describing the explicit and geometric construction ofthe limit point by iterating the destabilizing modification construction, we considervarious aspects of the resulting stratification. For example, we conjecture that thestratification is nested, i.e. smaller strata are contained in the closures of biggerones. This can be shown for bundles of rank 2. A calculation in deformation theoryshows that the set Lq ⊂ MDR of points (V,∇) such that limt→0(V, t∇) = q, isa lagrangian subspace for the natural symplectic form. We conjecture that thesesubspaces are closed and form a nice foliation 7.4. We mention in §6 that the sametheory works for the parabolic or orbifold cases, and point out a new phenomenonthere: the biggest open generic stratum no longer necessarily corresponds to unitarybundles. The Hodge type of the generic stratum varies with the choice of parabolicweights, with constancy over polytopes. At the end of the paper we do sometheoretical work (which was missing from [54]) necessary for proving the existenceof limit points in the case of principal bundles. All along the way, we try to identifynatural questions for further study.

It is a great pleasure to dedicate this paper to Professor Ramanan. I would liketo thank him for the numerous conversations we have had over the years, startingfrom my time as a graduate student, in which he explained his insightful points of

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DESTABILIZING MODIFICATIONS 3

view on everything connected to vector bundles. These ideas are infused throughoutthe paper.

I would also like to thank Jonathan Aidan, Daniel Bertrand, Philip Boalch,David Dumas, Jaya Iyer, Ludmil Katzarkov, Bruno Klingler, Vladimir Kostov,Anatoly Libgober, Ania Otwinowska, Tony Pantev, Claude Sabbah, and SzilardSzabo for interesting communications related to the subjects of this paper.

2. Griffiths transverse filtrations

Suppose X is a smooth projective curve, and V is a vector bundle with inte-grable holomorphic connection∇ : V → V ⊗OX

Ω1X . A Griffiths transverse filtration

is a decreasing filtration of V by strict subbundles

V = F 0 ⊃ F 1 ⊃ F 2 · · · ⊃ F k = 0

which satisfies the Griffiths transversality condition

∇ : F p → F p−1 ⊗OXΩ1

X .

In this case put E := GrF (V ) :=⊕

p Ep with Ep := F p/F p+1. Define the OX -

linear map

θ : Ep → Ep−1 ⊗OXΩ1

X

using ∇. Precisely, if e is a section of Ep, lift it to a section f of F p and notethat by the transversality condition, ∇f is a section of F p−1 ⊗OX

Ω1X . Define

θ(e) to be the projection of ∇(f) into Ep−1 ⊗OXΩ1

X . Again by the transversalitycondition, θ(e) is independent of the choice of lifting f . If a is a section of OX then∇(af) = a∇(f)+ f ⊗ da but the second term projects to zero in Ep−1 ⊗OX

Ω1X , so

θ(ae) = aθ(e), that is θ is OX -linear.We call (E, θ) the associated-graded Higgs bundle corresponding to (V,∇, F •).Griffiths-transverse filtrations are the first main piece of structure of variations

of Hodge structure, and in that context the map θ is known as the “Kodaira-Spencermap”. This kind of filtration of a bundle with connection was generalized to thenotion of “good filtration” for D-modules, and has appeared in many places.

A complex variation of Hodge structure consists of a (V,∇, F •) such that fur-thermore there exists a ∇-flat hermitian complex form which is nondegenerate oneach piece of the filtration, and with a certain alternating positivity property (if wesplit the filtration by an orthogonal decomposition, then the form should have sign(−1)p on the piece splitting F p/F p+1). For a VHS, the associated-graded Higgsbundle (E, θ) is semistable.

The historical variation of Hodge structure picture is motivation for consideringthe filtrations and Kodaira-Spencer maps, however we don’t use the polarizationwhich is not a complex holomorphic object. Instead, we concentrate on the semista-bility condition.

Definition 2.1. We say that (V,∇, F •) is gr-semistable (resp. gr-stable) ifthe associated-graded Higgs bundle (E, θ) is semistable (resp. stable) as a Higgsbundle.

The Higgs bundle (E, θ) is a fixed point of the C∗ action, which is equivalent

to saying that we have a structure of system of Hodge bundles [52, 4.1], i.e. adecomposition E =

⊕p E

p with θ : Ep → Ep−1 ⊗OXΩ1

X .

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4 C. SIMPSON

Remark 2.2. If (E, θ) is a system of Hodge bundles, then it is semistable asa Higgs bundle if and only if it is semistable as a system of Hodge bundles. Inparticular, if it is not a semistable Higgs bundle then the maximal destabilizingsubobject H ⊂ E is itself a system of Hodge bundles, that is H =

⊕Hp with

Hp := H ∩Ep. Indeed, if (E, θ) is not semistable, it is easy to see by uniqueness ofthe maximal destabilizing subsheaf that H must be preserved by the C∗ action.

Such objects have appeared in geometric Langlands theory under the name“opers”:

Example 2.3. An oper is a vector bundle with integrable connection and Grif-fiths transverse filtration (V,∇, F •) such that F • is a full flag, i.e. Ep = GrpF (V )are line bundles, and

θ : Ep ∼=→ Ep−1 ⊗OXΩ1

X

are isomorphisms. If g ≥ 1 then an oper is gr-semistable.

This motivates the following definition and terminology.

Definition 2.4. A partial oper is a vector bundle with integrable connectionand Griffiths-transverse filtration (V,∇, F •) which is gr-semistable.

Every integrable connection supports at least one partial oper structure.

Theorem 2.5. Suppose (V,∇) is a vector bundle with integrable connection ona smooth projective curve X. Then there exists a gr-semistable Griffiths-transversefiltration giving a partial oper structure (V,∇, F •).

The proof will be given in the next section.

3. Construction of a gr-semistable filtration

If (E =⊕

Ep, θ) is a system of Hodge bundles, for any k let E[k] denote thesystem of Hodge bundles with Hodge index shifted, so that

(E[k])p := Ep−k.

Let (V,∇) be fixed. Suppose we are given a Griffiths-transverse filtration F •

such that (GrF (V ), θ) is not a semistable Higgs bundle. Choose H to be the max-imal destabilizing subobject, which is a sub-system of Hodge bundles of GrF (V ).Thus

H =⊕

Hp, Hp ⊂ GrpF (V ) = F pV/F p+1V.

Note that the Hp are strict subbundles here. We can consider

Hp−1 ⊂ V/F pV

which is again a strict subbundle.Define a new filtration G• of V by

(3.1) Gp := ker

(V → V/F pV

Hp−1

).

The condition θ(Hp) ⊂ Hp−1 ⊗OXΩ1

X means that the new filtration G• is againGriffiths-transverse. We have exact sequences

0 → GrpF (V )/Hp → GrpG(V ) → Hp−1 → 0

186


which, added all together, can be written as an exact sequence of systems of Hodgebundles

(3.2) 0 → GrF (V )/H → GrG(V ) → H [1] → 0.

We would like to show that the process of starting with a filtration F • andreplacing it with the modified filtration G• stops after a finite number of steps, ata gr-semistable filtration. As long as the result is still not gr-semistable, we canchoose a maximal destabilizing subobject and continue. To show that the processstops, we will define a collection of invariants which decrease in lexicographic order.

For a system of Hodge bundles E, let β(E) denote the slope of the maximaldestabilizing subobject. Let ρ(E) denote the rank of the maximal destabilizingsubobject. Define the center of gravity to be

ζ(E) :=

∑rk(Ep) · prk(E)

.

This measures the average location of the Hodge indexing. In particular, supposeU = E[k]. Then Up = Ep−k so

ζ(U) =

∑prk(Up)

rk(U)=

∑prk(Ep−k)

rk(E)=

∑(p+ k)rk(Ep)

rk(E)= ζ(E) + k.

This gives the formula

(3.3) ζ(E[k]) = ζ(E) + k.

Now for any non-semistable system of Hodge bundles E, let H denote the maximaldestabilizing subobject and put

γ(E) := ζ(E/H)− ζ(H).

This normalizes things so that γ(E [k]) = γ(E).Denote βF := β(GrF (V ), θ), ρF := ρ(GrF (V ), θ), and γF := γ(GrF (V ), θ).

Lemma 3.1. In the process F • �→ G• described above, and assuming that G•

is also not gr-semistable, then the triple of invariants (β, ρ, γ) decreases strictlyin the lexicographic ordering. In other words, (βG, ρG, γG) is strictly smaller than(βF , ρF , γF ).

Proof. Use the exact sequence (3.2) and the formula (3.3). If K denotes themaximal destabilizing subobject of GrG(V ) then by (3.2), slope(K) ≤ slope(H).

If equality then the map Ku→ H [1] is injective, otherwise the kernel would be

semistable of the same slope and couldn’t map to GrF (V )/H. Thus rk(K) ≤rk(H), and again in case of equality then u would be an isomorphism, thereforeζ(K) = ζ(H) + 1 by 3.3 but ζ(GrG(V )/K) = ζ(GrF (V )/H) so γG < γF . �

In order to show that the γ(E) remain bounded, observe the following.

Lemma 3.2. If (V,∇) is an irreducible connection, and F • is a Griffiths-transverse filtration, then there are no gaps in the Ep = GrpF (V ), that is there

are no p′ < p < p′′ such that Ep = 0 but Ep′ �= 0 and Ep′′ �= 0.

Proof. If there were such a gap, then by Griffiths transversality the piece F p =F p+1 would be a nontrivial subbundle preserved by the connection, contradictingirreducibility of (V,∇). �

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6 C. SIMPSON

Lemma 3.3. Suppose (V,∇) is an irreducible connection. For all E = GrF (V )coming from non-gr-semistable Griffiths-transverse filtrations of our fixed (V,∇),each of the invariants β(E), ρ(E) and γ(E) can take on only finitely many values.

Proof. For the slope and rank this is clear. For γ(E) it follows from Lemma3.2. �

Proof of Theorem 2.5. Assume first of all that (V,∇) is irreducible. Underthe operation F • �→ G•, the triple of invariants (β, ρ, γ) (which takes only finitelymany values by Lemma 3.3) decreases strictly in the lexicographic ordering byLemma 3.1 until we get to a gr-semistable filtration.

For a general (V,∇), glue together the filtrations provided by the previousparagraph on the semisimple subquotients of its Jordan-Holder filtration. This canbe done after possibly shifting the indexing of the filtrations. �

Question 3.4. Understand what would happen if we tried to do the aboveprocedure in the case dim(X) ≥ 2 where the destabilizing subobjects could be torsion-free sheaves but not reflexive.

4. Interpretation in terms of the nonabelian Hodge filtration

Consider the “nonabelian Hodge filtration” moduli space

MHod(X, r) = {(λ, V,∇), ∇ : V → V ⊗ Ω1X , ∇(ae) = a∇(e) + λd(a)e}

with its map λ : MHod(X, r) → A1, such that:

—λ−1(0) = MH is the Hitchin moduli space of semistable Higgs bundles of rank rand degree 0; and—λ−1(1) = MDR is the moduli space of integrable connections of rank r.

The group Gm acts on MHod over its action on A1, via the formula t·(λ, V,∇) =

(tλ, V, t∇). Therefore all of the fixed points have to lie over λ = 0, that is they arein MH . We can write

(MH)Gm =⋃α

Pα

as a union of connected pieces. There is a unique piece denoted P0 along which allGm-orbits of MH are incoming. As discussed in §5.4 it corresponds to the moduli ofsemistable bundles with θ = 0. For the other pieces, certain invariants such as theranks and degrees of the Hodge bundles serve to distinguish components, howeverthe decomposition into connected components Pα could be finer.

Lemma 4.1. For any y ∈ MHod, the limit limt→0 t · y exists, and is in oneof the Pα. The limit is the associated-graded Higgs bundle of any gr-semistableGriffiths-transverse filtration on the connection corresponding to y.

Proof. An abstract proof was given in [54]. For λ(y) �= 0 in which case wemay assume λ(y) = 1 i.e. y ∈ MDR, the convergence can also be viewed as acorollary of Theorem 2.5. Indeed, y corresponds to a vector bundle with integrableconnection (V,∇) and if we choose a gr-semistable Griffiths-transverse filtration F •

then the limit can be calculated as

limλ→0

(V, λ∇) = (GrF (V ), θ).

This can be seen as follows. The Rees construction gives a locally free sheaf

ξ(V, F ) :=∑

λ−pF pV ⊗OX×A1 ⊂ V ⊗OX×Gm.

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overX×A1 and by Griffiths transversality the product λ∇ extends to a λ-connectionon ξ(V, F ) in the X-direction (here λ denotes the coordinate on A1). This familyprovides a morphism A

1 → MHod compatible with the Gm-action and having limitpoint

(ξ(V, F ), λ∇)|λ=0 = (GrF (V ), θ).

If λ(y) = 0 i.e. y ∈ MH then a construction similar to that of Theorem 2.5 givesa calculation of the limit. Alternatively, on the moduli space of Higgs bundles it iseasy to see from the properness of the Hitchin map [28] that the limit exists. �

We next note that the limit is unique.

Corollary 4.2. If F • and G• are two gr-semistable filtrations for the same(V,∇) then the Higgs bundles (GrF (V ), θF ) and (GrG(V ), θG) are S-equivalent,that is the associated-graded polystable objects corresponding to their Jordan-Holderfiltrations, are isomorphic.

Proof. The moduli space is a separated scheme whose points correspond toS-equivalence classes of objects. �

Now, given that the limiting Higgs bundle is unique, we can use it to measurewhether the partial oper structure will be unique or not:

Proposition 4.3. Suppose (V,∇) is a vector bundle with integrable connectionand let (E, θ) be the unique polystable Higgs bundle in the S-equivalence class ofthe limit. Then the gr-semistable Griffiths transverse filtration for (V,∇) is uniqueup to translation of indices, if and only if (E, θ) is stable.

Proof. Consider the Rees families (ξ(V, F ), λ∇) and (ξ(V,G), λ∇) on X×A1

constructed for 4.1 above. These are vector bundles with ΛHod-connections onX × A1 (see [53, I, p. 87], [54]). For any t ∈ A1 the space of morphisms from(ξ(V, F ), λ∇)t to (ξ(V,G), λ∇)t is the zero-th hypercohomology on X × {t} of acomplex on X×A

1 flat over OA1 , obtained by the Rees construction applied to thede Rham complex of V ∗ ⊗ V using the filtrations F and G. As is classically well-known, see [42] for example, there is a complex of finite rank bundles over A1 whosefiber over t calculates the space of morphisms, so the dimension is semicontinuousin t. For t �= 0 the dimension is at least 1, containing the identity of (V,∇). Ifthe unique limiting Higgs bundle (E, θ) is stable then the dimension of the spaceof morphisms at t = 0 is equal to 1, showing that the dimension is always 1. Itfollows that the Hom between these, relative to the base A1, is a rank one locallyfree sheaf over A

1 with action of Gm, and this relative Hom commutes with basechange. After appropriately shifting one of the filtrations we get a Gm-invariantsection which translates back to equality of the filtrations.

On the other hand, if the limiting Higgs bundle is not stable, we can choosea sub-system of Hodge bundles and apply the construction (3.1) to change thefiltration. The exact sequence (3.2) shows that the new filtration is different fromthe old. �

5. The oper stratification

As is generally the case for a Gm-action, the map y �→ limt→0 t · y is a con-structible map from MHod to the fixed point set

⋃α Pα.

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8 C. SIMPSON

Proposition 5.1. For any α, the subset Gα ⊂ MDR(X, r) consisting of allpoints y such that limt→0 t · y ∈ Pα is locally closed. These partition the modulispace into the oper stratification

MDR(X, r) =⋃α

Gα.

Furthermore, for any point p ∈ Pα (which corresponds to an S-equivalence class ofsystems of Hodge bundles), the set Lp ⊂ MDR(X, r) of points y with limt→0 t·y = p,is a locally closed subscheme (given its reduced subscheme structure).

Proof. This is the classical Bialynicki-Birula theory [31]: the moduli spacecan be embedded Gm-equivariantly in P

N with a linear action; the stratification andfibrations are induced by those of PN but with refinement of the Pα into connectedcomponents. �

For the moduli of Higgs bundles we have a similar stratification with strata

denoted Gα ⊂ MH(X, r) again defined as the sets of points such that limt→0 t · y ∈Pα.

5.1. Opers. The uniformizing Higgs bundles [28] are of the form E = E1 ⊕E0 = L⊕ L′ a direct sum of two line bundles with E0 = L′ ∼= L⊗K−1

X , such thatθ : E1 → E0 ⊗Ω1

X is an isomorphism. The space of these is connected, determinedby the choice of L ∈ Picg−1(X).

For bundles of rank r, one of the connected components Pα is the space ofsystems of Hodge bundles of the form E0 ⊕ · · · ⊕Er where the Ei are line bundles,

with θ : Ei∼=→ Ei−1 ⊗ Ω1

X . Any such E is isomorphic to a symmetric powerE ∼= Symr−1(F ) for F = L ⊕ L′ of rank 2 as above. These systems of Hodgebundles are rigid up to tensoring with a line bundle, indeed once E0 is chosen theremaining pieces are determined by Ei = E0⊗K⊗i

X . Thus, the set of them forms aconnected component Pα where α is just a notation for its indexing element. Thedeterminant map Pα → Pic0(X) is finite.

The classical moduli space of GL(r)-opers [5] [20] is the subset Gα defined inProposition 5.1 corresponding to the space Pα of symmetric powers of uniformizingHiggs bundles.

The stratum of classical opers Gα is closed, because the corresponding stratum

Gα is closed in MH. It also has minimal dimension among the strata, as can beseen from Lemma 7.3 below. We conjecture that it is the unique closed stratumand the unique stratum of minimal dimension. These are easy to see in the case ofrank 2, see §7.3.

5.2. Variations of Hodge structure. If (V,∇, F •, 〈·, ·〉) is a polarized com-plex VHS then the underlying Hodge filtration F • (which is Griffiths-transverse bydefinition) is gr-semistable. The Higgs bundle (GrF (V ), θ) is the one which cor-responds to (V,∇) by the nonabelian Hodge correspondence. This implies that if(V,∇) is a VHS with irreducible monodromy representation then it is gr-stable. Inthis case the filtration F • is unique and the process of iterating the destabilizingmodification described in §3 provides a construction of the Hodge filtration startingfrom just the bundle with its connection (V,∇).

For any stratum Gα as in Proposition 5.1, let GVHSα ⊂ MDR(X, r) be the real

analytic moduli space of polarized complex variations of Hodge structure whose

190


underlying filtered bundle is in Gα. We have a diagram of real analytic varieties

GVHSα

real↪→ Gα

∼= ↘ ↙Pα

.

Under the nonabelian Hodge identification ν : MDR(X, r) ∼= MH(X, r) the spaceGVHS

α of variations of Hodge structure is equal to Pα and the diagonal isomorphismin the above diagram is the identity when viewed in this way.

The other points of Gα don’t necessarily correspond to points of Gα under thenonabelian Hodge identification ν, and indeed it seems reasonable to make the

Conjecture 5.2.GVHS

α = Gα ∩ ν−1(Gα).

In a similar vein, let MB(X, r)R denote the real subspace of representationswhich go into some possibly indefinite unitary group U(p, q).

Lemma 5.3. Restricting to the subset of smooth points, GVHSα is a connected

component of Gα ∩MB(X, r)R.

The proof will be given in §7.1 below.On the other hand, it is easy to see that there are other connected components

too, for example when p, q > 0 a general representation π1 → U(p, q) will stillcorrespond to a stable vector bundle, so it gives a point in G0 ∩MB(X, r)R whichis not a unitary representation. These points probably correspond to twistor-likesections of Hitchin’s twistor space, but which don’t correspond to preferred sections.

Question 5.4. How can we distinguish GVHSα among the connected components

of Gα ∩MB(X, r)R?

5.3. Families of VHS. We now explain one possible motivation for lookingat the stratification by the Gα, related to Lemma 5.3.

Suppose given a smooth projective family of curves f : X → Y over a basewhich is allowed to be quasiprojective. Let Xy := f−1(y) be the fiber over a pointy ∈ Y . Then π1(Y, y) acts on MB(Xy, r). The fixed points are the representa-tions which come from global representations on the total space X of the fibration(approximately, up to considerations involving the center of the group and so on).In the de Rham point of view there is a “connection” on the relative de Rhammoduli space MDR(X/Y, r) → Y . This can be called the “nonabelian Gauss-Maninconnection” but is also known as the system of “isomonodromic deformation” equa-tions, or Painleve VI for the universal family of 4-pointed P1’s. The fixed points ofthe above action on the Betti space correspond to global horizontal sections of then.a.GM connection. These have been studied by many authors.

The global horizontal sections will often be rigid (hence VHS’s) and in any casecan be deformed, as horizontal sections, to VHS’s. Let ρ(y) ∈ MB(Xy, r) denote aglobal horizontal section which is globally a VHS; then we will have

ρ(y) ∈ GVHSα (Xy)

for all y in a neighborhood of an initial point y0. The combinatorial data corre-sponding to the stratum α will be invariant as y moves; let’s assume that we cansay that the index sets of our stratifications remain locally constant as a function ofy, and that ρ(y) stays in the “same stratum”—although that would require further

191

10 C. SIMPSON

proof. Then we have, in particular, ρ(y) ∈ Gα(y). The stratum Gα(y) will notbe fully invariant by the n.a.GM connection [26], so ρ(y) has to lie in the subsetof points which, when displaced by the n.a.GM connection, remain in the same(i.e. corresponding) stratum. We get a system of equations constraining the pointρ(y). One can hope that in certain special cases, these equations have only isolatedpoints as solutions in each fiber Gα(y). A program for finding examples of globalhorizontal sections ρ would be to look for these isolated points and then verify ifthe family of such points is horizontal.

5.4. The open stratum. At the other end of the range of possible dimensionsof strata, is the unique open stratum G0 in each component of the moduli space.When X is a smooth compact curve of genus ≥ 2, the open stratum consists ofconnections of the form (V, ∂ + A) where V is a polystable vector bundle, ∂ isthe unique flat unitary connection, and A ∈ H0(End(V ) ⊗ Ω1

X). The Griffiths-transverse filtration is trivial, and the corresponding stratum of systems of Hodgebundles is P0 = U(X, r), the moduli space of semistable vector bundles on X. The

Higgs stratum is just the cotangent bundle G0 = T ∗P0 and G0 is a principal T ∗P0-torsor over P0. The space GVHS

0 is just the space of unitary representations. Thesituation becomes more interesting when we look at parabolic bundles or bundleson an orbifold.

6. The parabolic or orbifold cases

Let (X, {x1, . . . , xk}) be a curve with some marked points, and fix semisimpleunitary conjugacy classes C1, . . . , Ck ⊂ GL(r). We can consider the various modulispaces of representations, Higgs bundles, connections, or λ-connections on U :=X − {x1, . . . , xk} with logarithmic structure at the points xi and correspondingmonodromies contained in the conjugacy classes Ci respectively. These objectscorrespond to parabolic vector bundles with real parabolic weights and λ-connection∇ respecting the parabolic filtration and inducing the appropriate multiple of theidentity on each graded piece. At λ = 0 the residues of the Higgs field are zerosince we are assuming that the conjugacy classes are unitary, whereas for λ �= 0 therelationship between parabolic weights, residue and conjugacy classes is as describedin [51] [41].

If in addition the conjugacy classes are assumed to be of finite order, the para-bolic weights should be rational, and our objects may then be viewed as lying on anorbicurve or Deligne-Mumford stack X ′ with ramification orders mi correspondingto the common denominators of the weights at xi [7] [13] [30]. Denote by

MH(X′;C1, . . . , Ck) ⊂ MHod(X

′;C1, . . . , Ck) ⊃ MDR(X′;C1, . . . , Ck) etc

the resulting moduli spaces. One could further assume for simplicity that thereexists a global projective etale Galois covering Z → X ′ with Galois group G. Themap Z → X is a ramified Galois covering such that the numbers of branches inthe ramification points above xi are always equal to mi. Local systems or otherobjects on X ′ are the same thing as G-equivariant objects on Z. This enables aneasy construction of the moduli stacks and spaces. More general constructions ofmoduli spaces of parabolic objects can be found in [58] [57] [43] [37] [32] [29] [15].

The correspondence between Higgs bundles and local systems works in thiscase, as can easily be seen by pulling back to the covering Z (although the analysiscan also be done directly). All of the other related structures also work the same

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way. The C∗ action preserves the conjugacy classes since they are assumed to beunitary. There is a proper Hitchin fibration on the space of Higgs bundles [19][58] [37], and similarly the nonabelian Hodge filtration satisfies the positive weightproperty saying that limits exist as in Lemma 4.1.

A first classical case appearing already in the paper of Narasimhan and Seshadri[45] is when there is a single parabolic point x1 and the residual conjugacy classof the connection is that of the scalar matrix d/r. Orbifold bundles of this typecorrespond to usual bundles on X ′ of degree d, which can have projectively flatconnections. This led to moduli spaces in which all semistable points are stablewhen (r, d) = 1.

The semistable ⇒ stable phenomenon occurs fairly generally in the orbifold orparabolic setting. The Cj have to satisfy the equation

∏det(Cj) = 1. However,

if the eigenvalues of the individual blocks are chosen sufficiently generally, it oftenhappens that there is no sub-collection of 0 < r′ < r eigenvalues at each point,such that the product over all points is equal to 1. We call this Kostov’s genericitycondition. It implies that all representations are automatically irreducible, or interms of vector bundles with connection or Higgs bundles, semistable objects areautomatically stable.

This situation is particularly relevant for our present discussion:

Lemma 6.1. Suppose the conjugacy classes C1, . . . , Ck satisfy Kostov’s generic-ity condition, then any vector bundle with connection (V,∇) ∈ MDR(X

′;C1, . . . , Ck)has a unique partial oper structure, the filtration being unique up to shifting the in-dices. Also, in this case the moduli stacks M•(X

′;C1, . . . , Ck) are smooth, and theyare Gm-gerbs over the corresponding moduli spaces M•(X

′;C1, . . . , Ck).

Proof. If all semistable objects are automatically stable, then the same is truefor the associated-graded Higgs bundles. Proposition 4.3, which works equally wellin the orbifold case, implies that the filtration is unique up to a shift.

For the statement about moduli stacks, stable objects admit only scalar au-tomorphisms, and the GIT construction of the moduli spaces of stable objectsyields etale-locally fine moduli, which is enough to see the gerb statement. Forsmoothness, the obstructions land in the trace-free part of H2 which vanishes byduality. �

Note that when there is a single point and the conjugacy class is scalar multi-plication by a primitive r-th root of unity, this is Kostov-generic, and we are exactlyin the original situation considered by Narasimhan and Seshadri corresponding tobundles of degree coprime to the rank.

Corollary 6.2. If the conjugacy classes satisfy Kostov’s genericity condi-tion, then MHod(X

′;C1, . . . , Ck) is smooth over A1, and the fixed point sets of the

Gm-action Pα are smooth. The projections Gα → Pα and Gα → Pα are smoothfibrations topologically equivalent to the normal bundle N

˜Gα/Pα→ Pα.

Proof. The fixed-point sets are smooth by the theory of Bialynicki-Birula.Let GHod

α ⊂ MHod(X′;C1, . . . , Ck) denote the subset of points whose limit lies in

Pα. It is smooth, and the fiber over λ = 0 is the smooth Gα. By the action ofGm, the map GHod

α → A1 is smooth everywhere. Let GHodα,V HS ⊂ GHod

α denotethe subspace of points which correspond to variations of Hodge structure. The

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12 C. SIMPSON

preferred-section trivialization gives

GHodα,V HS

∼= Pα × A1

compatibly with the Gm-action.Let T ⊂ GHod

α be a tubular neighborhood of GHodα,V HS, and denote by T0 the

fiber over λ = 0. We can suppose that ∂T0 is transverse to the vector field definedby the action of R∗

>0 ⊂ Gm. Consequently, there is ε such that ∂T is transverse tothis vector field for any |λ| < 2ε.

Choose a trivialization F : ∂T0 × A1 ∼= ∂T compatible with the map to A1.Then define a map

R : R∗>0 × ∂T0 → Gα,

R(t, x) := t · F (x, t−1).

This is a diffeomorphism in the region t > ε−1/2. On the other hand, let Tε be thefiber over λ = ε and consider the map ε−1 : Tε → Gα. This glues in with the map Rdefined on the region t ≥ ε−1 to give a topological trivialization of Gα. Everythingcan be done relative to Pα so we get a homeomorphism between Gα → Pα and the

normal bundle of Pα in Gα. The same happens for Gα → Pα. �

6.1. The Hodge type of the open stratum. The open stratum consistsof variations of Hodge structure of a certain type. This type can change as wemove the conjugacy classes. For example, the set of vectors of conjugacy classes(C1, . . . , Ck) for k points in P1, for which there exists a unitary representation, isdefined by some inequalities on the logarithms of the eigenvalues [1] [6] [8] [15][56].

If we fix a collection of partitions at each point, then the space of unitaryconjugacy classes corresponding to those partitions is a real simplex, and the setof finite-order conjugacy classes is the set of rational points therein. The set ofnon-Kostov-generic points is a union of hyperplanes, as can be seen from the iden-tification between residues and parabolic weights, so the set of Kostov-generic pointsdecomposes into a union of open chambers bounded by pieces of hyperplanes (inparticular, the chambers are polytopes).

Suppose that we know that the varieties M•(X′;C1, . . . , Ck) are connected—

this is the case, for example, if at least one of the conjugacy classes has distincteigenvalues [33]. Then there is a single open stratum. If the genericity condition issatisfied so all objects are stable, then the partial oper structure at a general pointis unique, and its Hodge type (up to translation) depends only on the conjugacyclasses. Furthermore, given a Kostov-generic collection of unitary conjugacy classes(C1, . . . , Ck) and a parabolic system of Hodge bundles with these conjugacy classes,if we perturb slightly the parabolic weights then the object remains stable. Thisshows that the Hodge type for the unique open stratum is constant on the Kostov-generic chambers.

A limit argument should show that if we fix a particular Hodge type {hp,q},then the set of vectors of conjugacy classes (C1, . . . , Ck) for which the points inthe open stratum admit a partial oper structure with given hp,q, is closed. Wedon’t do that proof here, though—let’s just assume it’s true. It certainly holdsfor the unitary case h0,0 = r. This closed set is then a closed polytope. For theunitary case, it is the polytope found and studied by Boden, Hu, Yokogawa [14][15], Biswas [8] [9], Agnihotri, Woodward [1], Belkale [6]. The boundary is a unionof pieces of the non-Kostov-generic hyperplanes.

194


The fact that the Hodge type varies in a way which is locally constant overpolytopes fits into the general philosophy which comes out of the results of Libgober[36] and Budur [17]. There is a direct relationship in the rigid case: by Katz’salgorithm the unique point in a rigid moduli space can be expressed motivically,and the Hodge type is locally constant on polytopes by the theory of [36] [17].

There seems to be a further interesting phenomenon going on at the boundaryof these polytopes: the complex variations of Hodge structure corresponding to thefixed point limits of points in the open stratum, become automatically reducible.In the unitary case this is pointed out in several places in the references (see forexample Remark (4) after Theorem 7 on page 75 of [6], or also the last phrase ofTheorem 3.23 of [9]): irreducible unitary representations exist only in the interiorof the polytope. A sketch of proof in general is to say that if there is a stable systemof Hodge bundles with given parabolic structure then the parabolic structure canbe perturbed keeping stability and keeping the same Hodge type (it is the sameargument as was used above at the interior points of the chambers, which is indeedessentially the same as in the references).

In other words, at the walls between the chambers, points in the open stratumbecome gr-semistable but not gr-stable. Now, the non-uniqueness of the partialoper structure, Proposition 4.3, is exactly what allows the Hodge type to jump.

For generic (C1, . . . , Ck) where the moduli space is smooth, the space of complexvariations of Hodge structure corresponding to the open stratum is a real form ofthe Betti moduli space (extend to the parabolic case the proof of Lemma 5.3 in §7.1below). It is defined algebraically in MB(X;C1, . . . , Ck) as a connected componentof the space of representations into some U(p, q).

However, at boundary points (C1, . . . , Ck) between the different Hodge poly-topes this subspace is concentrated on the singular locus of reducible representations(whereas in most cases there still exist irreducible representations). The open stra-tum is presented as a conic fibration over a fixed point set inside the singular locus.The reader will be convinced that this kind of thing can happen by looking at thecase of the real circle x2+y2 = t: for t > 0 it is a real form of the complex quadraticcurve, but at t = 0 the real points are just the singularity of two crossing complexlines. It is an interesting question to understand what the degeneration of the realform of the smooth space looks like for the case of MB(X;C1, . . . , Ck).

7. Deformation theory

The deformation theory follows [11] [16] [39] and others. The discussion ex-tends without further mention to the case where X is an orbifold. Further workis needed [59] [16] for more general parabolic cases corresponding for example tonon-unitary conjugacy classes of local monodromy operators.

Suppose (V,∇) is an irreducible connection. Consider the complex

End(V )⊗ Ω•X := [End(V )

d→ End(V )⊗ Ω1X ].

The space of infinitesimal deformations, or the tangent space to the moduli stackMDR at (V,∇), is

Def(V,∇) = H1(End(V )⊗ Ω•

X).

Given a Griffiths-transverse filtration F • we get a decreasing filtration of the com-plex End(V )⊗ Ω•

X defined by

F p(End(V )) := {ϕ ∈ End(V ), ϕ : F qV → F p+qV },

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14 C. SIMPSON

F p(End(V )⊗ Ω1X) := F p−1(End(V ))⊗ Ω1

X .

The F p(End(V )⊗ Ω•X) are subcomplexes, so we can take the spectral sequence of

the filtered complex (End(V )⊗ Ω•X , F •). It induces a filtration F •Def(V,∇) and

the spectral sequence is

(7.1) Hi(GrpF (End(V )⊗ Ω•

X)) = Ep,i−p1 ⇒ GrpFH

i(End(V )⊗ Ω•X).

The obstruction theory is controlled by the H2 of the trace-free part, so if (V,∇)

is irreducible then the deformations are unobstructed, there are no problems withextra automorphisms, and the moduli space is smooth.

The limiting system of Hodge bundles (GrF (V ), θ) has its own deformationtheory in the world of Higgs bundles. The complex End(GrF (V ))⊗Ω•

X is made asabove, but the differentials using θ are OX -linear maps. The tangent space to themoduli stack MH at (GrF (V ), θ) is calculated as

Def(GrF (V ), θ) = H1(End(GrF (V ))⊗ Ω•

X).

The complex End(GrF (V ))⊗Ω•X has a direct sum decomposition, and indeed it is

the associated-graded of the previous one:

End(GrF (V ))⊗ Ω•X = GrF (End(V )⊗ Ω•

X).

This tells us that the E•,•1 term of the previous spectral sequence corresponds to

the deformation theory of (GrF (V ), θ).

Lemma 7.1. If (GrF (V ), θ) is stable then the spectral sequence (7.1) degeneratesat E1 with

(7.2) GrpF (Def(V,∇)) = H1(GrpF (End(V ))

θ→ Grp−1F (End(V ))⊗ Ω•

X

).

Let T (Gα/Pα) denote the relative tangent bundle of the fibration Gα → Pα where itis smooth. Then the two middle subspaces of the filtration at p = 0, 1 are interpretedin a geometric way as

F 0Def(V,∇) = T (Gα)(V,∇), and F 1Def(V,∇) = T (Gα/Pα)(V,∇).

Proof. Since (GrF (V ), θ) is stable,

H0(GrF (End(V )⊗ Ω•

X)) = C, H2(GrF (End(V )⊗ Ω•

X)) = C

and these are the same as at the limit of the spectral sequence. By invarianceof the Euler characteristic the same must be true for H

1 so the spectral sequencedegenerates. The geometric interpretation may be seen by looking at a cocycledescription: F 0Def(V,∇) is the space of deformations of the bundle with itsfiltration (V, F •,∇). Following out 4.1, this is the tangent space to Gα; whileF 1Def(V,∇) is the space of deformations of (V, F •,∇) inducing the trivial de-formation on the associated-graded so it is the tangent space to the fiber of theprojection Gα → Pα. �

7.1. At a variation of Hodge structure. Suppose (V,∇) ∈ GVHSα is an

irreducible complex variation of Hodge structure. Then End(V ) is a real VHS ofweight 0, independent of the scalar choice of polarization on V . The hypercoho-mology Def(V,∇) = H1(End(V )⊗Ω•

X) is a real Hodge structure of weight 1, withthe same Hodge filtration as defined above and which we write as

HR ⊂ Def(V,∇) ∼=⊕

Hk,1−k.

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This decomposition splits the Hodge filtration and is naturally isomorphic to thedecomposition (7.2). The real subbundle of End(V ) is defined to consist of matriceswhich infinitesimally preserve the polarization form, so the real subspace HR is thespace of deformations of the monodromy representation globally preserving thepolarization form, i.e. as a representation in the real group U(p, q). In other wordsit is the tangent space to MB(X, r)R in the notations of Lemma 5.3.

Proof of Lemma 5.3. Both GVHSα and Gα ∩MB(X, r)R have the same tan-

gent spaces at (V,∇). The tangent space of Gα is the sum F 0 =⊕

k≥0Hk,1−k.

The intersection of F 0 with HR is (H1,0 ⊕ H0,1)R which is the tangent space of

GVHSα . �

The symmetry of Hodge numbers extends to other points too:

Corollary 7.2. For any gr-stable point (V,∇), not necessarily a VHS, let F •

be the unique (up to shift) partial oper structure. Then the induced filtration on thetangent space of the moduli stack at (V,∇) satisfies

dimGrpFDef(V,∇) = dimGr1−pF Def(V,∇).

Proof. Let (V ′,∇′) denote the VHS corresponding to the system of Hodgebundles (GrF (V ), θ). Note that (GrF (V

′), θ′) ∼= (GrF (V ), θ), so

GrpFDef(V,∇) ∼= H1(End(GrF (V ))⊗ Ω•

X) ∼= GrpFDef(V ′,∇′).

Therefore we can write GrpFDef(V,∇) ∼= Hp,1−p(End(V ′)⊗Ω•

X), and these spacesare associated to a real Hodge structure of weight one, so we get the claimedsymmetry of dimensions. �

In the Hitchin moduli space the tangent space at a point (E, θ) ∈ Pα decom-poses as a direct sum under the Gm-action, and this decomposition is compatiblewith the filtration, so it gives a splitting:

T (MH)(E,θ)∼=

⊕GrpFT (MH)(E,θ)

∼=⊕

Hp,1−p.

The tangent space to the fixed point set is Gr0FT (MH)(E,θ) = T (Pα)(E,θ), whereasthe pieces p < 0 may be identified as the “outgoing” directions for the Gm-action.

If Pα = P0 is the lowest piece corresponding to the open stratum G0 ⊂ M thenthere are no outgoing directions, so the terms of Hodge type (p, 1 − p) vanish forp < 0. By symmetry they vanish for p > 1, which says that GrpFT (MH)(E,θ) = 0unless p = 0, 1; the same holds for MDR. Furthermore, the two pieces are dual bythe symplectic form (see below), so we can identify

H0,1 = Gr0FT (MH)(E,θ)

∼= T (P0)(E,θ),

H1,0 = Gr1FT (MH)(E,θ)

∼= T ∗(P0)(E,θ).

We even have an isomorphism of fibrations

G0 T ∗P0

↓ ∼= ↓P0 P0,

whereas G0 → P0 is a twisted form of the same fibration (it is a torsor). Theseobservations are elementary and classical when X is a smooth projective curve,however they also hold in the parabolic or orbifold case when the weights aregeneric so that all points are gr-stable.

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16 C. SIMPSON

This includes cases where there are no stable parabolic vector bundles, but thelowest stratum P0 corresponds to variations of Hodge structure of nontrivial Hodgetype.

7.2. Lagrangian property for the fibers of the projections Gα → Pα.There is a natural symplectic form on the tangent bundle to the moduli space overthe open subset of stable points, given in cohomological terms [11, §4] as the cupproduct followed by Lie bracket

H1(End(V )⊗ Ω•

X)⊗H1(End(V )⊗ Ω•

X)[ , ]◦∪−→ H

2(End(V )⊗ Ω•X) = C.

In the moduli space of connections on a smooth projective curve, the openstratum G0 fibers over the moduli space P0 of semistable vector bundles. In the

Hitchin moduli space, under the isomorphism G0∼= T ∗P0, the symplectic form is

equal to the standard one [11], in particular the fibers are lagrangian.It has been noticed by several authors [24] [16] [55] [35] [2] that the fibers of

the projection G0 → P0 (over stable points) are similarly lagrangian subspaces ofthe moduli space of connections.

We point out that the lagrangian property extends to the fibers in all strata,at gr-stable points.

Lemma 7.3. Suppose p = (E, θ) ∈ Pα is a stable system of Hodge bundles.Then the fiber Lp of the projection Gα → Pα over p is a lagrangian subspace ofMDR.

Proof. Fix a point (V,∇) which is gr-stable, hence also stable itself. ByLemma 7.1, the tangent space to the fiber of Gα/Pα is the subspace F 1H1 ⊂Def(V,∇).

In degree two, F 1H2(End(V )⊗Ω•X) is the full hypercohomology space H2 = C

whereas F 2H2 = 0. On the other hand, the Hodge filtration is compatible with cupproduct, so

∪ : F 1H

1 ⊗ F 1H

1 → F 2H

2 = 0,

which is to say that F 1H

1 ⊂ Def(V,∇) is an isotropic subspace for the symplecticform. The symmetry property Corollary 7.2 readily implies that F 1H1 has half thedimension, so it is lagrangian. �

A natural problem is to understand the relationship between the lagrangianfibers on different strata.

Question 7.4. Do the lagrangian fibers of the projections fit together into asmooth lagrangian foliation with closed leaves?

This is of course true within a given stratum Gα; does it remain true as αvaries?

A heuristic argument for why this might be the case is that if a fiber of Gα → Pα

were not closed in MDR, this might lead to a projective curve in MDR which cannotexist since MB is affine.

A similar closedness or properness property was proven in [55] for the case ofFuchsian equations.

On the other hand, the analogous statement for MH is not true: the fibers of

Gα → Pα definitely do sometimes have nontrivial closures in MH, indeed any fibercontained in the compact nilpotent cone will be non-closed.

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If the answer to Question 7.4 is affirmative, it could be useful for the contextof geometric Langlands, where the full moduli stack BunGL(r) might profitably bereplaced by the algebraic space of leaves of the foliation (see Arinkin’s work [3]).They share the open set of semistable bundles. This could help in the ramified case[23]—when there are no semistable bundles, it would seem logical to consider thelowest stratum P0 parametrizing variations of Hodge structure.

7.3. Nestedness of the stratifications. We are given a stratification ordisjoint decomposition into locally closed subsets M =

∐α Gα. Say that it is

nested if there is a partial order on the index set such that

Gα −Gα =∐β<α

Gβ.

If this is the case, the partial order is defined by the condition

β ≤ α ⇔ Gβ ⊂ Gα.

The arrangement of the strata is the partially ordered index set.

Conjecture 7.5. The stratifications of MDR(X, r) and MH(X, r) defined inProposition 5.1 are nested, and the arrangements of their strata are the same via theidentification of each stratum with the corresponding fixed point set in MH(X, r).

The arrangement of the strata in the Hitchin moduli space has been studiedin [10], and for rank 3 the connected components of the fixed point sets have beenclassified in [21].

Theorem 7.6. Conjecture 7.5 is true for bundles of rank r = 2 on a smoothprojective curve X of genus g ≥ 2.

Proof. Hitchin [28] identifies explicitly the connected components Pe of thefixed point set MH(X, 2)Gm . These are indexed by an integer 0 ≤ e ≤ g − 1, whereP0 is the space of semistable vector bundles and for e > 1, Pe is the space of systemsof Hodge bundles which are direct sums of line bundles of degrees e and −e. Adeformation theory argument will show that starting with a point in Ge we candeform it to a family of λ-connections which go into the next stratum Ge−1.

At e = 0, the space P0 is the moduli space of rank two semistable vector bundleson X, which is sometimes denoted UX(2), and is known to be an irreducible variety.

For e > 0 the space Pe parametrizes Higgs bundles of the form

E = E0 ⊕ E1, θ : E1 → E0 ⊗ Ω1X ,

where E0 and E1 are line bundles of degrees −e and e respectively. We requiree > 0 because a Higgs bundle of the above form with deg(Ei) = e = 0 would besemistable but not stable, in the same S-equivalence class as the polystable vectorbundle E0 ⊕ E1 which is a point in P0.

The map θ is a section of the line bundle (E1)∗⊗E0⊗Ω1X of degree 2g−2−2e, so

e ≤ g−1 and in the case of equality, θ is an isomorphism. LetD be the divisor of θ; itis an effective divisor of degree 2g−2−2e, and the space of such is Sym2g−2−2e(X).Given D and E1 ∈ Pice(X) then E is determined by E0 = E1 ⊗ TX ⊗ OX(D).Thus, Pe

∼= Sym2g−2−2e(X)× Pice(X) which is a smooth irreducible variety. Allpoints are stable.

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18 C. SIMPSON

The strata of the oper stratification are enumerated as Ge ⊂ MDR(X, 2) for

0 ≤ e ≤ g − 1; the corresponding strata of MH(X, 2) are denoted Ge. The upper(smallest) stratum e = g − 1 is the case of classical opers.

To show the nested property, it suffices to show that Ge ⊂ Ge−1 for e ≥ 2. Notethat for e = 1 this is automatic since G0 is an open dense subset so G0 = MDR.We will consider a point (V,∇) ∈ Ge and deform it to a family (Vt,∇t) ∈ Ge−1 fort �= 0, with limit (V,∇) as t → 0.

Having (V,∇, F •) ∈ Ge means that there is an exact sequence

(7.3) 0 → E1 → V → E0 → 0

with E1 = F 1 and E0 = F 0/F 1. The limit point in Pe is E = E1 ⊕ E0 withθ := ∇ : E1 → E0 ⊗ Ω1

X . Choose a point p ∈ X and let L := E1(−p). Letϕ : L → V be the inclusion. Consider the functor of deformations of (V,∇, L, ϕ).It is controlled by the hypercohomology of the complex

C• := End(V )⊕Hom(L,L) → End(V )⊗ Ω1X ⊕Hom(L, V )

where the differential is the matrix(∇ 0

− ◦ ϕ ϕ ◦ −

).

There is a long exact sequence

. . . → Hi(Hom(L,L) → Hom(L, V )) → H

i(C•) → Hi(End(V )⊗ Ω•

X) → . . . .

The fact that our filtration is gr-stable implies that the spectral sequence as-sociated to the filtered complex (End(V )⊗Ω•

X , F •) degenerates at E•,•1 . The VHS

on End(GrF (V )) has Hodge types (1,−1) + (0, 0) + (−1, 1) so its H1 has types(2,−1) + . . .+ (−1, 2). The associated-graded of H1(End(V )⊗ Ω•

X) has pieces

Gr2FH1 = H0(Hom(E0, E1)⊗ Ω1

X),

Gr1FH1 = H

1(Hom(E0, E1) → (Hom(E0, E0)⊕Hom(E1, E1))⊗ Ω1X),

Gr0FH1 = H

1(Hom(E1, E1)⊕Hom(E0, E0) → Hom(E1, E0)⊗ Ω1X),

Gr−1F H

1 = H1(Hom(E1, E0)).

The degeneration gives a surjection H1 →→ Gr−1

F H1, in other words

H1(End(V )⊗ Ω•

X) → H1(Hom(E1, E0)) → 0.

The definition L = E1(−p) gives an exact sequence of the form

(7.4) 0 → Hom(E1, E0) → Hom(L,E0) → Cp → 0.

Note that deg(L) > 0 but deg(E0) < 0 so H0(Hom(L,E0)) = 0, hence the longexact sequence for (7.4) becomes

0 → C → H1(Hom(E1, E0)) → H1(Hom(L,E0)) → 0.

Putting these two together we conclude that the map

H1(End(V )⊗ Ω•

X) → H1(Hom(L,E0))

is surjective, but there is an element in its kernel which maps to something nonzeroin H1(Hom(E1, E0)). Another exact sequence like (7.4) gives a surjection

H1(OX) = H1(Hom(L,L)) → H1(Hom(L,E1)) → 0.

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The long exact sequence for Hom from L to the short exact sequence (7.3) gives

H1(Hom(L,E1)) → H1(Hom(L, V )) → H1(Hom(L,E0)) → 0.

Therefore the map

(7.5) H1(Hom(L,L))⊕H1(End(V )⊗ Ω•

X) → H1(Hom(L, V ))

is surjective but not an isomorphism. This gives vanishing of the obstruction theoryfor deforming (V,∇, L, ϕ). And the map (7.5) has an element in its kernel whichmaps to something nonzero in H1(Hom(E1, E0)). Hence there is a deformation of(V,∇, L, ϕ) which doesn’t extend to a deformation of E1 ⊂ V . A one-parameterfamily of (Vt,∇t, Lt, ϕt) thus gives the deformation from (V,∇) into the next loweststratum Ge−1 where Lt of degree e− 1 will be the destabilizing subsheaf. �

8. Principal objects

Ramanan would want us to consider also principal bundles for arbitrary reduc-tive structure group G. Notice that the category of partial oper structures, whileposessing tensor product and dual operations, is not tannakian because morphismsof filtered objects need not be strict. The passage to principal objects should bedone by hand. One can construct the various moduli spaces of principal bundleswith connection, principal Higgs bundles, and the nonabelian Hodge moduli spaceof principal bundles with λ-connection

MH(X,G) ⊂ MHod(X,G) ⊃ MDR(X,G)↓ ↓ ↓{0} ⊂ A1 ⊃ {1}

.

In this section we indicate how to prove that the limit limt→0 t · p exists for anyp ∈ MHod(X,G). The discussion using harmonic bundles will be more technicalthan the previous sections of the paper.

Embedding G ↪→ GL(r), it suffices to show that the map of moduli spacesMHod(X,G) → MHod(X,GL(r)) is finite and then apply Lemma 4.1. Such a finite-ness statement was considered for MH and MDR in [53] along with the constructionof the homeomorphism between these two spaces. The same statements were men-tionned for MHod in [54], however the discussion there was inadequate. The mainissue is to prove that the map from the moduli space of framed harmonic bundlesto the GIT moduli space MHod(X,G) is proper. The distinct arguments at λ = 0and λ = 1 given in [53] don’t immediately generalize to intermediate values of λ.This is somewhat similar to—and inspired by—the convergence questions treatedrecently by Mochizuki in [41]. We give an argument based on the topology of themoduli space together with its Hitchin map; it would be interesting to have a moredirect argument with explicit estimates.

Fix a basepoint x ∈ X, and let RHod(X,x,G) be the parameter variety for(λ, P,∇, ζ) where λ ∈ A

1, P is a principal G-bundle with λ-connection ∇ (suchthat (P,∇) is semistable with vanishing Chern classes) and ζ : G ∼= Px is a framefor P at the point x. The group G acts and

RHod(X,x,G) → MHod(X,G)

is a universal categorical quotient. Indeed, RHod(X,x,G) is constructed first as aclosed subscheme of RHod(X,x,GL(r)) for a closed embedding G ⊂ GL(r) by thesame construction as in [53, II, §9], which treated the Higgs case. This was based

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20 C. SIMPSON

on [53, I,§4] which treats the general Λ-module case so the discussion transposesto MHod. One can choose a GL(r)-linearized line bundle on RHod(X,x,GL(r))for which every point is semistable, which by restriction gives a G-linearized linebundle on RHod(X,x,G) for which every point is semistable. Mumford’s theorythen gives the universal categorical quotient.

Fix a compact real form J ⊂ G. For any λ ∈ A1 a harmonic metric on aprincipal bundle with λ-connection (P,∇) is a C∞ reduction of structure group h ⊂P from G to J satisfying some equations (see [41] for example) which interpolatebetween the Yang-Mills-Higgs equations at λ = 0 and the harmonic map equationsat λ = 1. Let RJ

Hod(X,x,G) ⊂ RHod(X,x,G) denote the subset of (λ, P,∇, ζ) suchthat there exists a harmonic metric h compatible with the frame ζ at x, in otherwords ζ : J → hx. This condition fixes h uniquely.

Lemma 8.1. The map RJHod(X,x,G) → MHod(X,G) is proper, and induces

homeomorphisms

MH(X,G)× A1 ∼= RJ

Hod(X,x,G)/J ∼= MHod(X,G).

Proof. The moduli space of harmonic λ-connections has a natural productstructure RJ

Hod(X,x,G) ∼= HarJ(X,G) × A1 where HarJ(X,G) is the space of

framed harmonic G-bundles; and the topological quotient by the action of J onthe framing is again a product (HarJ (X,G)/J)×A1. The second homeomorphismfollows from the properness statement, by the discussion in [53]. There the proper-ness was proven at λ = 0, 1. At λ = 0 we get HarJ(X,G)/J ∼= MH(X,G), whichgives the first homeomorphism. At λ = 1 we get HarJ(X,G)/J ∼= MDR(X,G).

Fix G ⊂ GL(r) such that J ⊂ U(r), then RJHod(X,x,G) is a closed subset of

RU(r)Hod (X,x,GL(r)). Using this, one can show that the lemma for GL(r) implies

the lemma for G.Suppose now G = GL(r), with compact subgroup J = U(r). The map on the

subset of stable points RJHod(X,x, r)s → MHod(X, r)s is proper, using the fact that

MHod(X, r)s is a fine moduli space, plus the main estimate for the constructionof Hermitian-Einstein harmonic metric solutions by the method of Donaldson’sfunctional. This estimate is explained for the case of λ-connections in [41].

Given a polystable point (λ, V,∇) ∈ MHod(X, r) we can associate a polystableHiggs bundle (E, θ) in the preferred section corresponding to the harmonic bundleassociated to any harmonic metric on (V,∇). The (E, θ) is unique up to iso-morphism, in particular the value of the Hitchin map Ψ = det(θ − t) ∈ CN iswell-defined. This gives a set-theoretically defined map Ψ : MHod(X, r) → CN .For a sequence of points ρi ∈ RJ

Hod(X,x, r), there is a convergent subsequence ifand only if the sequence Ψ[ρi] contains a bounded subsequence. Hence, in orderto prove properness of the map RJ

Hod(X,x, r) → MHod(X, r) it suffices to provethat the function Ψ is locally bounded on MHod(X, r). This is obvious on the fiberλ = 0 where the Hitchin map Ψ is an algebraic map. On the fiber λ = 1, an ar-gument using the characterization of harmonic metrics as ones which minimize theenergy ‖θ‖2L2 again shows that Ψ is locally bounded. In particular, the lemma holdsover λ = 0 and λ = 1, indeed this was the proof of [53] for the homeomorphismMH(X, r) ∼= MDR(X, r).

Properness over the set of stable points means that the map MHod(X, r)s →MH(X, r)s is continuous, hence Ψ is continuous over MHod(X, r)s. Similarly, itis continuous on any stratum obtained by fixing the type of the decomposition of

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a polystable object into isotypical components. A corollary is that for any point[(λ, V,∇)] ∈ MHod(X, r), the function C∗ � t �→ Ψ(tλ, V, t∇) is continuous.

Suppose we have a sequence of points pi → q converging in MHod(X, r), butwhere |Ψ(pi)| → ∞. Assume that they are all in the same fiber Mλ over a fixedvalue 0 �= λ ∈ A1. The points λ−1pi converge to λ−1q in MDR(X, r), so (by theenergy argument referred to above) we have a bound |Ψ(λ−1pi)| ≤ C1. Fix a curvesegment γ ⊂ C joining λ−1 to 1 but not passing through 0. The function t �→ Ψ(tq)is continuous by the previous paragraph, so there is a bound |Ψ(tq)| ≤ C2 for t ∈ γ.On the other hand, again by the continuity of the previous paragraph, for anyconstant C > C1 there exists a sequence of points ti ∈ γ such that |Ψ(tipi)| = C.Possibly going to a subsequence, we can assume that tipi → q′ as a limit of harmonicbundles. Continuity of the Hitchin map on MH says that |Ψ(q′)| = C. The mapfrom the space of harmonic bundles to MHod is continuous so the limit tipi → q′

also holds in MHod. On the other hand, we can assume ti → t in γ (again possiblyafter going to a subsequence), which gives tipi → tq in MHod. Separatedness of thescheme MHod implies that the topological space is Hausdorff, so tq = q′. If C > C2

this contradicts the bound |Ψ(tq)| ≤ C2 for t ∈ γ. We obtain a contradiction to theassumption |Ψ(pi)| → ∞, so we have proven that |Ψ(pi)| is locally bounded.

In the fiber over each fixed λ ∈ A1, this shows properness, hence the homeo-morphism statements, hence that Ψ is continuous. Now using the connectednessand separatedness properties of MHod, an argument similar to that of the previ-ous paragraph will allow us to show boundedness of Ψ globally over MHod withoutrestricting to a single fiber. Suppose pi → q in MHod over a convergent sequenceλi → λ ∈ A1, but with |Ψ(pi)| → ∞. Fix a preferred section σ : A1 → MHod and wemay assume that σ(λi) is connected to pi by a path γi : [0, 1] → Mλi

. If necessaryreplacing X by a sufficiently high genus covering, we can view MHod as a family ofconnected normal varieties. Thus we can assume that the paths γi converge to apath γ connecting q to σ(λ) in the fiber Mλ. Fix a constant C > supt|Ψ(γ(t))|, inparticular also C > |Ψ(σ(λi))|, indeed the Ψ(σ(λi)) are all the same because σ wasa preferred section. For large values of i we have

|Ψ(σ(λi))| = |Ψ(γi(0))| < C < |Ψ(γi(1))| = |Ψ(pi)|.Continuity of Ψ in each fiber Mλi

shows that there are ti ∈ [0, 1] with |Ψ(γi(ti))| =C. Going to a subsequence we get convergence of the harmonic bundles associatedto the points γi(ti), keeping the same norm of the Hitchin map. Thus γi(ti) →q′ with |Ψ(q′)| = C. For a further subsequence, ti → t and as in the previousargument, separatedness of the moduli space implies that q′ = γ(t), contradictingthe choice of C. This proves that Ψ is locally bounded, which in turn impliesproperness of the map in the first statement of the lemma for GL(r), to completethe proof. �

Suppose now given an injective group homomorphism between reductive groupsG ↪→ H. Choose compact real forms J ⊂ G and K ⊂ H such that the homomor-phism is compatible: J → H. This gives a diagram

RJHod(X,x,G) → RK

Hod(X,x,H)↓ ↓

MHod(X,G) → MHod(X,H)

where the vertical maps are proper by the previous lemma. The upper horizontalmap is a closed embedding, indeed we can choose H ↪→ GL(r) which also induces

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22 C. SIMPSON

G ↪→ GL(r), and the schemes RHod(X,x,G) and RHod(X,x,H) are by constructionclosed subschemes of RHod(X,x,GL(r)) [53]. The subsets of framings compatiblewith the harmonic metrics are closed, so the upper horizontal map is an inclusioncompatible with closed embeddings into RHod(X,x,GL(r)), hence it is a closedembedding. In particular, it is proper. This implies that the bottom map is proper.

Corollary 8.2. Given a group homomorphism with finite kernel between re-ductive groups G → H the resulting map on moduli spaces

MHod(X,G) → MHod(X,H)

is finite.

Proof. If G → H is injective, the above argument shows that the map onmoduli spaces is proper. It is quasi-finite [53] so it is finite. Then the same argumentas in [53] yields the same statement in the case of a group homomorphism withfinite kernel between reductive groups. �

Corollary 8.3. Suppose G is a reductive group. Then for any point p ∈MHod(X,G) the limit point limt→0 t · p exists and is unique in the fixed point setMH(X,G)Gm. Hence we get a stratification of MDR(X,G) just as in Proposition5.1.

Proof. Choose G ↪→ GL(r); apply Lemma 4.1 together with the finiteness ofCorollary 8.2 to get existence of the limit. Uniqueness follows from separatednessof the moduli space [53]. The stratification is defined in the same way as in 5.1. �

The above arguments prove that the limit points exist, however it would begood to have a geometric construction analogous to what we did in §2, §3. Thisshould involve a principal-bundle approach to the instability flag [48].

Question 8.4. How to give an explicit description of the limiting points interms of Griffiths-transverse parabolic reductions in the case of principal G-bundles?

We obtain the oper stratification of MDR(X,G) just as in Proposition 5.1. Thesmallest stratum consisting of G-opers is treated in much detail in [5].

It would be good to generalize the other elements of our discussion §§5, 6, 7to the principal bundle case too. The theory of parabolic structures would hit thesame complications mentionned by Seshadri in the present conference. The theoryof deformations should follow [11].

References

[1] S. Agnihotri, C. Woodward, Eigenvalues of products of unitary matrices and quantum Schu-bert calculus, Math. Res. Lett. 5 (1998), 817–836.

[2] J. Aidan, Doctoral dissertation, preliminary version (June 2008).[3] D. Arinkin, Orthogonality of natural sheaves on moduli stacks of SL(2)-bundles with con-

nections on P1 minus 4 points. Selecta Mathematica 7 (2001), 213–239.[4] V. Balaji, I. Biswas, D. Nagaraj, Principal bundles over projective manifolds with parabolic

structure over a divisor, Tohoku Math. J. 53 (2001), 337–367.[5] A. Beilinson, V. Drinfeld, Opers, Preprint arXiv:math/0501398.[6] P. Belkale, Local systems on P1 − S for S a finite set, Compositio Math. 129 (2001), 67–86.[7] I. Biswas, Parabolic bundles as orbifold bundles, Duke Math. J., 88 (1997), 305–325.[8] I. Biswas, A criterion for the existence of a parabolic stable bundle of rank 2 over the pro-

jective plane, Int. J. Math. 9 (1998), 523–533.[9] I. Biswas, On the existence of unitary flat connections over the punctured sphere with given

local monodromy around the puncture, Asian J. Math. 3 (1999), 333–344.

204


[10] I. Biswas, T. Gomez, A Torelli theorem for the moduli space of Higgs bundles on a curve,The Quarterly Journal of Mathematics 54 (2003), 159–169.

[11] I. Biswas, S. Ramanan, An Infinitesimal study of the moduli of Hitchin pairs, Journal of theLondon Math. Soc. 49 (1994), 219–231.

[12] P. Boalch, From Klein to Painleve via Fourier, Laplace and Jimbo, Proc. London Math. Soc.90 (2005), 167–208.

[13] H. Boden, Representations of orbifold groups and parabolic bundles, Comm. Math. Helv. 66

(1991), 389–447.[14] H. Boden, Y. Hu, Variations of moduli of parabolic bundles, Math. Ann. 301 (1995), 539–559.[15] H. Boden, K. Yokogawa, Moduli spaces of parabolic Higgs bundles and parabolic K(D) pairs

over smooth curves, I, Internat. J. Math. 7 (1996), 573–598.[16] F. Bottacin, Symplectic geometry on moduli spaces of stable pairs, Ann. Sci. Ecole Norm.

Sup. 28 (1995) 391–433.[17] N. Budur, Unitary local systems, multiplier ideals, and polynomial periodicity of Hodge num-

bers, Preprint arXiv:math/0610382.[18] K. Corlette, C. Simpson, On the classification of rank-two representations of quasiprojective

fundamental groups, Compositio Math. 144 (2008), 1271–1331.[19] R. Donagi, E. Markman, Spectral covers, algebraically completely integrable, hamiltonian sys-

tems, and moduli of bundles, Integrable Systems and Quantum Groups (Montecatini Terme1993), Springer L.N.M. 1620 (1996), 1–119.

[20] E. Frenkel, Wakimoto modules, opers and the center at the critical level, Adv. in Math. 195(2005), 297–404.

[21] O. Garcia-Prada, P. Gothen, V. Munoz, Betti numbers of the moduli space of rank 3 parabolicHiggs bundles, Memoirs Amer. Math. Soc. 187 (2007).

[22] O. Garcia-Prada, S. Ramanan, Twisted Higgs bundles and the fundamental group of compactKahler manifolds, Math. Res. Lett. 7 (2000), 517–535.

[23] S. Gukov, E. Witten, Gauge Theory, ramification, and the geometric Langlands program,Preprint arXiv:hep-th/0612073v1.

[24] R. Hain, On a generalizaton of Hilbert’s 21st problem, Ann. Sci. E.N.S. 19 (1986), 609–627.[25] T. Hausel, Compactification of moduli of Higgs bundles, J. Reine Angew. Math. 503 (1998),

169–192.[26] V. Heu-Berlinger, Deformations isomonodromiques des connexions de rang 2 sur les courbes,

Doctoral dissertation, Rennes (2008).[27] N. Hitchin, Stable bundles and integrable systems, Duke Math. J. 54 (1987), 91–114.[28] N. Hitchin, The self-duality equations on a Riemann surface, Proc. London Math. Soc. (3)

55 (1987), 59–126.[29] M. Inaba, K. Iwasaki, Masa-Hiko Saito, Moduli of stable parabolic connections, Riemann-

Hilbert correspondence and geometry of Painleve equation of type VI, Part I, Publ. Res. Inst.Math. Sci. 42 (2006), 987–1089.

[30] J. Iyer, C. Simpson, A relation between the parabolic Chern characters of the de Rhambundles, Math. Ann. 338 (2007), 347–383.

[31] F. Kirwan, Cohomology of quotients in symplectic and algebraic geometry, Princeton Uni-versity Press (1984).

[32] H. Konno, Construction of the moduli space of stable parabolic Higgs bundles on a Riemannsurface, J. Math. Soc. Japan 45 (1993), 253–276.

[33] V. Kostov, The connectedness of some varieties and the Deligne-Simpson problem, J. ofDynamical and Control Systems 11 (2005), 125–155.

[34] V. Kostov, The Deligne-Simpson problem for zero index of rigidity, Perspectives In ComplexAnalysis, Differential Geometry And Mathematical Physics, St Constantin (Bulgaria), 2000,World Scientific (2001), 1–20.

[35] S. Lekaus, Unipotent flat bundles and Higgs bundles over compact Kahler manifolds, Trans.Amer. Math. Soc. 357 (2005) 4647–4659.

[36] A. Libgober, Problems in topology of the complements to plane singular curves, Singularities

in geometry and topology, World Sci. Publ. (2007), 370–387.[37] M. Logares, J. Martens, Moduli of parabolic Higgs bundles and Atiyah algebroids, Preprint

math.AG/0811.0817.[38] F. X. Machu, Monodromy of a class of logarithmic connections on an elliptic curve, Symme-

try, Integrability and Geometry: Methods and Applications 3 (2007), 31pp.

205

24 C. SIMPSON

[39] E. Markman, Spectral Curves and Integrable Systems, Compositio Math. 93 (1994), 255–290.[40] M. Maruyama, K. Yokogawa, Moduli of parabolic stable sheaves, Math. Ann. 293 (1992),

77–99.[41] T. Mochizuki, Kobayashi-Hitchin correspondence for tame harmonic bundles II, Geometry

and Topology 13 (2009) 359–455.[42] D. Mumford, Abelian Varieties. Tata Institute of Fundamental Research Studies in Mathe-

matics (1970).

[43] H. Nakajima, Hyper-Kahler structures on moduli spaces of parabolic Higgs bundles on Rie-mann surfaces, Moduli of vector bundles (Sanda, Kyoto, 1994), Lect. Notes Pure Appl. Math.179 (1996), 199–208.

[44] M.S. Narasimhan, S. Ramanan, Deformations of the moduli space of vector bundles over analgebraic curve, Ann. Math. 101 (1975), 391–417.

[45] M.S. Narasimhan, C. Seshadri, Stable and unitary bundles on a compact Riemann surface,Ann. of Math. 82 (1965), 540–564.

[46] N. Nitsure, Moduli space of semistable pairs on a curve, Proc. London Math. Soc. 62 (1991),275–300.

[47] Nitin Nitsure, Moduli of semistable logarithmic connections, Journal of the A.M.S. 6 (1993),597–609.

[48] S.Ramanan, A.Ramanathan, Some remarks on the instability flag, Tohoku Math. J. 36(1984), 269–291.

[49] C. Seshadri, Moduli of π-vector bundles over an algebraic curve, Questions on AlgebraicVarieties (C.I.M.E., III Ciclo, Varenna, 1969), Edizioni Cremonese, Rome (1970) 139–260.

[50] A. Schmitt, Projective moduli for Hitchin pairs, Int. J. of Math. 1 (1998), 107–118.[51] C. Simpson, Harmonic bundles on noncompact curves. J.A.M.S. 3 (1990), 713–770.[52] C. Simpson, Higgs bundles and local systems, Publ. Math. I.H.E.S. 75 (1992), 5–95.[53] C. Simpson, Moduli of representations of the fundamental group of a smooth projective va-

riety. I,II, Publ. Math. I.H.E.S. 79, 80 (1994, 1995).[54] C. Simpson, The Hodge filtration on nonabelian cohomology, Algebraic geometry—Santa

Cruz 1995, , Proc. Sympos. Pure Math., 62, Part 2, A.M.S. (1997), 217–281.[55] S. Szabo, Deformations of Fuchsian equations and logarithmic connections, Preprint

math.AG/0703230.[56] C. Teleman, C. Woodward, Parabolic bundles, products of conjugacy classes and Gromov-

Witten invariants, Ann. Inst. Fourier 53 (2003), 713–748.[57] M. Thaddeus, Variation of moduli of parabolic Higgs bundles, J. Reine Angew. Math. 547

(2002), 1–14.[58] K. Yokogawa, Compactification of moduli of parabolic sheaves and moduli of parabolic Higgs

sheaves, J. Math. Kyoto Univ. 33 (1993), 451–504.[59] K. Yokogawa, Infinitesimal deformation of parabolic Higgs sheaves, Internat. J. Math 6

(1995), 125–148.

CNRS, Laboratoire J. A. Dieudonne, UMR 6621 Universite de Nice-Sophia Antipolis,

06108 Nice, Cedex 2, France

E-mail address: [email protected]: http://math.unice.fr/∼carlos/

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CONM/522www.ams.orgAMS on the Webwww.ams.org

This volume contains a collection of papers from the Conference on Vector Bundles held at Miraflores de la Sierra, Madrid, Spain on June 16–20, 2008, which honored S. Ramanan on his 70th birthday.

The main areas covered in this volume are vector bundles, parabolic bundles, abelian varieties, Hilbert schemes, contact structures, index theory, Hodge theory, and geometric invariant theory. Professor Ramanan has made important contributions in all of these areas.

vector bundles and complex geometry

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