m.j. bergvelt and j.m. rabin- supercurves, their jacobians, and super kp equations

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Vol. 98, No. 1 DUKE MATHEMATICAL JOURNAL © 1999

SUPERCURVES,THEIR JACOBIANS, AND SUPER KPEQUATIONS

M. J. BERGVELT and J. M. RABIN

Contents

1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22. Supercurves and their Jacobians . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

2.1. Supercurves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42.2. Duality and N = 2 curves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52.3. Super Riemann surfaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102.4. Integration on supercurves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112.5. Integration on the universal cover. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132.6. Sheaf cohomology for supercurves. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

2.7. Generic SKP curves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 162.8. Riemann bilinear relations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 172.9. The period map and cohomology. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 182.10. X as an extension of erX . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 12.11. X as an extension of êrX and symmetric period matrices . . . . . . . . 242.12. Moduli of invertible sheaves. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 252.13. Effective divisors and Poincaré sheaf for generic SKP curves. . . . . . . . . 272.14. Berezinian bundles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 292.15. Bundles on the Jacobian; θ-functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

3. Super Grassmannian, τ -function, and Baker function . . . . . . . . . . . . . . . . . . . . . 343.1. Super Grassmannians . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 343.2. The Chern class of Ber(gr) and the gl∞|∞ cocycle . . . . . . . . . . . . . . . . . 36

3.3. The Jacobian super Heisenberg algebra. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 383.4. Baker functions, the full super Heisenberg algebra, and τ -functions. . . . 39

4. The Krichever map and algebro-geometric solutions . . . . . . . . . . . . . . . . . . . . . . 424.1. The Krichever map . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 424.2. The Chern class of the Ber bundle on Pic0(X) . . . . . . . . . . . . . . . . . . . . . . . 4 44.3. Algebro-geometric τ - and Baker functions. . . . . . . . . . . . . . . . . . . . . . . . . . . 45

Appendix A. Duality and Serre duality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48A.1. Duality of - m o d u l e s . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 8A.2. Serre duality of supermanifolds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49

Appendix B. Real structures and conjugation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50

Received 10 January 1996.1991 Mathematics Subject Classification. Primary 14M30; Secondary 14H40, 32G20, 58F07,

17B65, 14H10, 32C11, 58A50.

1



2 BERGVELT AND RABIN

Appendix C. Linear equations in the super category . . . . . . . . . . . . . . . . . . . . . . . . . 50C . 1 . I n t r o d u c t i o n . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 0C.2. Quasideterminants . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51C.3. Restriction to superalgebra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52

C.4. Linear equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53Appendix D. Calculation of a super τ -function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53R e f e r e n c e s . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 5

1. Introduction. In this paper we study algebraic supercurves with a view towardapplications to super Kadomtsev-Petviashvili (SKP) hierarchies. We deal from thestart with supercurves X over a nonreduced ground ring ; that is, our curves carryglobal nilpotent constants. This has an advantage, compared to supercurves over thecomplex numbers C, that our curves can be nonsplit, but this comes at the price of some technical complications. The main problem is that the cohomology groups of coherent sheaves on our curves shouldbe thought of as finitely generatedmodules overthe ground ring , instead of vector spaces over C. In general these modules are of

course not free. Still we have in this situation Serre duality, as explained in AppendixA, the dualizing sheaf being the (relative) Berezinian sheaf erX. In applications toSKP there occurs a natural class of supercurves that we call generic SKP curves.For these curves, the most important sheaves—the structure sheaf and the dualizingsheaf—have free cohomology. In the latter part of the article we concentrate on thesecurves.

Supercurves exhibit a remarkable duality uncovered in [DRS]. The projectivizedcotangent bundle of any N = 1 supercurve has the structure of an N = 2 superRiemann surface (SRS), and supercurves come in dual pairs X,X whose associatedN = 2 SRSs coincide. Further, the (-valued) points of a supercurve can be identifiedwith the effective divisors of degree 1 on its dual. Ordinary N = 1 SRSs, widelystudied in the context of superstring theory, are self-dual under this duality. By theresulting identification of points with divisors they enjoy many of the properties thatdistinguish Riemann surfaces from higher-dimensional varieties. By exploiting theduality we extend this good behaviour to all supercurves.

In particular we define for all supercurves a contour integration for sections of erX, the holomorphic differentials in this situation. The endpoints of a supercontourturnouttobenot -points of our supercurve, but rather irreducible divisors, that is, -points on the dual curve! For SRSs these notions are the same, and our integration is ageneralization of the procedure already known for SRSs. We use this to prove Riemannbilinear relations, connecting in this situation periods of holomorphic differentials onour curve X with those on its dual curve.

In the case when the cohomology of the structure sheaf is free, for example, if X is a generic SKP curve, we can define a period matrix and use this to define theJacobian of X as the quotient of a super vector space by a lattice generated by the

period matrix. In this case the Jacobian is a smooth supermanifold. A key question is



SUPERCURVES 3

whether the Jacobian of a generic SKP curve admits ample line bundles (and henceembeddings in projective superspace) whose sections could serve as the super analogof θ-functions. We show that the symmetry of the even part of the period matrix(together with the automatic positivity of the imaginary part of the reduced matrix)

is sufficient for this, and we construct the super θ-functions in this case. We derivesome geometric necessary and sufficient conditions for this symmetry to hold, butit is not an automatic consequence of the Riemann bilinear period relations in thissuper context. Neither do we know an explicit example in which the symmetry fails.The usual proof that symmetry of the period matrix is necessary for existence of a(principal) polarization also fails, because crucial aspects of Hodge theory, particularlythe Hodge decomposition of cohomology, do not hold for supertori.

The motivation for writing this paper was our wish to generalize the theory of the algebro-geometric solutions to the KP hierarchy of nonlinear partial differentialequations, as described in [SW] and references therein, to the closest supersymmetricanalog, the “Jacobian” SKP hierarchy of Mulase and Rabin [Mu], [R1].

In the SKP case the geometric data leading to a solution include a supercurve X

and a line bundle with vanishing cohomology groups over X. For such a line bundleto exist, the supercurve X must have a structure sheaf X such that the associatedsplit sheaf X

split, obtained by putting the global nilpotent constants in equal tozero, is a direct sum X

split = redX ⊕, where red

X is the structure sheaf of theunderlying classical curve Xred and is an invertible red

X -sheaf of degree zero. Wecall such an X an SKP curve, and if moreover is not isomorphic to red

X we call X

a generic SKP curve.The Jacobian SKP hierarchy describes linear flows (t i ) on the Jacobian of X

(with even and odd flow parameters). The other known SKP hierarchies, of Maninand Radul [MR] and Kac and van de Leur [KL], describe flows on the universalJacobian over the moduli space of supercurves, in which X as well as vary with thet i [R1]. These are outside the scope of this paper, although we hope to return to themelsewhere. As in the nonsuper case, the basic objects in the theory are the (even and

odd) Baker functions, which are sections of (t i ) holomorphic except for a singlesimple pole, and a τ -function that is a section of the superdeterminant (Berezinian)bundle over a super Grassmanniangr. In contrast to the nonsuper case, we show thatthe Berezinian bundle has trivial Chern class, reflecting the fact that the Berezinian isa ratio of ordinary determinants. The super τ -function descends, essentially, to Jac(X)

as a section of a bundle with trivial Chern class also, and it can be expressed rationallyin terms of super θ-functions when these exist. (Its reduced part is a ratio of ordinaryτ -functions). We also obtain a formula for the even and odd Baker functions in termsof the τ -function, confirming that one must know the τ -function for the more generalKac–van de Leur flows to compute the Baker functions for even the Jacobian flowsin this way; see [DS], [T]. For this we need a slight extension of Cramer’s rule forsolving linear equations in even and odd variables, which is developed in an appendix

via the theory of quasideterminants. In another appendix we use the Baker functions




found in [R2] for Jacobian flow in the case of superelliptic curves to compute thecorresponding τ -function.

Among the problems remaining open we mention the following. (1) To obtain asharp criterion for when a super Jacobian admits ample line bundles, perhaps always?

(2) The fact that generic SKP curves have free cohomology is a helpful simplificationthat allows us to represent their period maps by matrices and results in their Jacobiansbeing smooth supermanifolds. However, our results should generalize to arbitrarysupercurves with more singular Jacobians. (3) To study the geometry of the universalJacobian and extend our analysis to the SKP system of Kac and van de Leur.

2. Supercurves and their Jacobians

2.1. Supercurves. Fix a Grassmann algebra over C; for instance, we could take = C[β1, β2, . . . , βn], the polynomial algebra generated by n odd indeterminates.Let (•, ) be the superscheme Spec , with underlying topological space a singlepoint •.

A smooth, compact, connected, complex supercurve over of dimension (1 | N) is

a pair (X,X), where X is a topological space and X is a sheaf of supercommutative-algebras over X, equipped with a structure morphism (X,X) → (•, ), such thatthe following conditions are met.

1. (X,redX ) is a smooth, compact, connected, complex curve, algebraic or holo-

morphic, depending on the category that one works in. Here redX is the reduced sheaf

of C-algebras on X obtained by quotienting out the nilpotents in the structure sheaf X.

2. For suitable open sets U α ⊂ X and suitable linearly independent odd elementsθ i

α of X(U α), we have

X(U α) = redX ⊗

θ 1

α , θ 2α , . . . , θ N

α

.

The U α’sabove are called coordinateneighborhoods of (X,X),and (zα , θ1α , θ 2

α , . . . ,

θ N α ) are called local coordinates for (X,X), if zα (mod nilpotents) is a local coordi-nate for (X,red

X ). On overlaps of coordinate neighborhoods U α ∩ U β we have

zβ = F βα

zα , θ j

α

,

θ iβ = i

βα

zα , θ j

α

.

(2.1)

Here the F βα are even functions and iβα odd ones, holomorphic or algebraic de-

pending on the category we are using.

Example 2.1.1. A special case is formed by the split supercurves. For N = 1 theyare given by transition functions

zβ = f βα (zα),

θβ = θαBβα (zα ),

(2.2)



SUPERCURVES 5

with f βα (zα), Bβα (zα ) even holomorphic (or algebraic) functions that are independentof the nilpotent constants in . So in this case the f βα are the transition functionsfor red

X and X = redX ⊗ | ⊗ , where is the red

X -module with transitionfunctions Bβα (zα ). Here and henceforth we denote by a vertical | a direct sum of free

-modules, with an evenly generated summand on the left and an odd one on theright.To any supercurve (X,X) there is canonically associated a split curve (X,X

split)

over C: just take Xsplit = X ⊗ /m = X/mX, with m = β1, . . . , βn the

maximal ideal of nilpotents in . There is a functor from the category of X-modulesto the category of X

split-modules that associates to a sheaf the associated split

sheaf split =/m.A -point of a supercurve (X,X) is a morphism φ : (•, ) → (X,X) such that

the composition with the structural morphism (X,X) → (•, ) is the identity (of (•, )). Locally, in an open set U α containing φ(•), a -point is given by speci-fying the images under the even -homomorphism φ : X(U α) → of the localcoordinates: pα = φ(zα), π i

α = φ(θ iα ). The local parameters (pα , π i

α ) of a -pointtransform precisely as the coordinates do; see (2.1). By quotienting out nilpotents in

a -point (pα, π iα), we obtain a complex number predα , the coordinate of the reduced

point of (X,redX ) corresponding to the -point (pα , π i

α ).

2.2. Duality and N = 2 curves. Our main interest is the theory of N = 1 super-curves, but as a valuable tool for the study of these curves we make use of N = 2curves as well in this paper. Indeed, as is well known (see [DRS], [Sc2]), one canassociate in a canonical way to an N = 1 curve an (untwisted) superconformal N = 2curve, as we now recall. The introduction of the superconformal N = 2 curve clarifiesthe whole theory of N = 1 supercurves.

Let from now on (X,X) be an N = 1 supercurve. Any invertible sheaf for(X,X) and any extension of by the structure sheaf

0 → X → → → 0

defines in the obvious way an N = 2 supercurve (X,). It has local coordinates(zα, θα, ρα), where (zα , θα ) are local coordinates for (X,X). On overlaps we have

zβ = F βα

zα, θα

,

θβ = βα

zα , θα

,

ρβ = H βα

zα, θα

ρα + φβα

zα , θα

.

(2.3)

(So H βα (zα, θα) is the transition function for the generators of the invertible sheaf .) We want to choose the extension (2.2) such that (X,) is superconformal, in thesense that the local differential form ωα = dzα − dθα ρα is globally defined up to ascale factor. Now

ωβ = dzβ − dθβ ρβ = dzα ∂F

∂zα −

∂

∂zα ρβ

− dθα

−

∂F

∂θα +

∂

∂θα ρβ

.




(Here we suppress the subscripts on F and , as we do below.) We see that for tobe superconformal we need

ρα =−

∂F

∂θα

+∂

∂θα

ρβ∂F

∂zα

−∂

∂zα

ρβ

or

ρβ =

∂F

∂θα

+∂F

∂zα

ρα

∂

∂θα

−∂

∂zα

ρα

.(2.4)

Conversely one checks that if (2.4) holds for all overlaps, the cocycle condition issatisfied, and that we obtain in this manner an N = 2 supercurve. To show that thissupercurve is an extension as in (2.2), it is useful to note that (2.4) can also be written

as

ρβ = ber

∂zF ∂z

∂θ F ∂θ

ρα +

∂θ F

∂θ .(2.5)

The homomorphism ber is defined in Appendix C; see (C.3). Recall that the localgenerators f α of the dualizing sheaf (see Appendix A) erX of (X,X) transform as

f β = ber

∂zF ∂z

∂θ F ∂θ

f α.(2.6)

If we denote by X the structure sheaf of the superconformal N = 2 supercurve just constructed, we see that we have an exact sequence

0 → X → X →erX → 0.(2.7)

X is the only extension of erX by the structure sheaf that is superconformal. Thissequence is trivial if it isomorphic (see [HS]) to a split sequence.

Definition 2.2.1. A supercurve is called projected if there is a cover of X suchthat the transition functions F βα in (2.1) are independent of the odd coordinates θ

jα .

For projected curves we have a projection morphism (X,X) → (X,redX ⊗ )

corresponding to the sheaf inclusion redX ⊗ → X. This inclusion can be defined

only for projected curves.A projected supercurve has a X that is a trivial extension but the converse is

not true, as we see when we discuss SRSs in Subsection 2.3. The relation betweenprojectedness of (X,X) and the triviality of the extension defining (X,X) is

discussed in detail in Subsection 2.11.



SUPERCURVES 7

Example 2.2.2. If (X,X) is split, (2.6) becomes

f β =∂zf βα

Bβα

f α.(2.8)

This means that in this case erX = −1 ⊗X = −1 ⊗ | ⊗ , where isthe canonical sheaf for red

X .Split curves are projected andthe sequence (2.7) becomes trivial. As anred

X -modulewe have X = (red

X ⊕) ⊗ | (⊕−1) ⊗ .

The map X →erX is locally described by the differential operator Dα

= ∂ρα .

Indeed, the operator Dα transforms homogeneously, D

β

= ber

∂zF ∂z∂θ F ∂θ

−1Dα, so

this defines a global (0 | 1)-dimensional distribution D, and the quotient of (X,X)

by this distribution is precisely (X,X).Now the distribution D annihilates the 1-form ω used to find X. This form

locally looks like ωα = dzα − dθαρα and its kernel is generated by Dα

and a second

operator Dα

= ∂θα +ρα ∂zα . (The operators that we call D and D are in the literature

also denoted by D+

and D−

; cf. [DRS].) To study the result of “quotienting by thedistribution D” we introduce in each coordinate neighborhood U α new coordinates:

zα = zα − θα ρα,

θα = θα ,

ρα = ρα .

In the sequel we drop the hat ˆ on θ and ρ, hopefully not causing too much confusion.In these new coordinates we have

Dα = ∂θα , Dα

= ∂ρα + θα ∂zα.

So the kernel of D consists locally of functions of zα , ρα . To see that this makesglobal sense we observe that

zβ = F (zα , ρα ) +DF (zα , ρα )

D(zα, ρα)(zα, ρα),

ρβ =DF (zα , ρα )

D(zα , ρα),

(2.9)

where D = ∂θ + θ ∂z. The details of this somewhat unpleasant calculation are left tothe reader. From (2.9) we see that X contains the structure sheaf X of anotherN = 1 supercurve: X is the sheaf of -algebras locally generated by zα , ρα . We callX = (X, X) the dual curve of (X,X). We have

0 → X → XD

−−→êrX → 0,(2.10)

whereêrX is the dualizing sheaf of the dual curve. One can show that the dual curve

of the dual curve is the original curve, thereby justifying the terminology.




Example 2.2.3. We continue the discussion of split curves. In this case (2.9) be-comes

zβ = f

zα

,

ρβ =∂

zf zα

B

zα

ρα,(2.11)

so the dual split curve is splitX = red

X ⊗ | −1 ⊗ . The Berezinian sheaf for thedual split curve has generators that satisfy

f β = B

zα

f α.(2.12)

This means that êrX =⊗ X =⊗ |⊗ .

A very useful geometric interpretation of the dual curve exists (see [DRS], [Sc2]):the points, (i.e., the -points) of the dual curve correspond precisely to the irreducibledivisors of the original curve and vice versa, as we presently discuss. In Subsection

2.4 we see that irreducible divisors are the limits that occur in contour integration ona supercurve.An irreducible divisor (for X) is locally given by an even function P α = zα − zα −

θα ρα ∈ X(U α ), where zα and ρα are now, respectively, even and odd constants, thatis, elements of . Two divisors P α , P β defined on coordinate neighborhoods U α , U β ,respectively, are said to correspond to each other on the overlap if

P β

zβ , θβ

= P α

zα , θα

g

zα , θα

, g

zα , θα

∈ ×

X,ev

U α ∩ U β

.(2.13)

(If R is a ring—or sheaf of rings—R× is the set of invertible elements.)

Lemma 2.2.4. Let (U,(U)) be a (1 | 1)-dimensional superdomain with coordi-

nates (z,θ) , and let f(z,θ) ∈ (U ). Then, with D = ∂θ + θ ∂z ,

f(z,θ) =

z − z − θρ

g(z,θ) ⇔ f

z, ρ

= 0, Df

z, ρ

= 0,

for g(z,θ) in (U ).

Applying Lemma 2.2.4 to (2.13) we find

P β

F (zα , ρα ),(zα, ρα)

= F

zα, ρα

− zβ −

zα , ρα

ρβ = 0,

DP β

F (zα , ρα ),(zα, ρα)

= DF

zα , ρα

− D

zα , ρα

ρβ = 0.

From this one sees that the parameters (zα , ρα) in the local expression for an irre-ducible divisor transform as in (2.9), so they are -points of the dual curve.

The N = 2 superconformal supercurve canonically associated to a supercurve hasa structure sheaf X that comes equipped with two sheaf maps D and D whose

kernels are the structure sheaves X and X of the original supercurve and its dual.



SUPERCURVES 9

The intersection of the kernels is the constant sheaf . The images of these mapsare the dualizing sheaves erX and êrX. In fact we can restrict D, D to thesubsheaves X, X, respectively, without changing the images. This gives us exactsequences

0 → →XD

−→êrX → 0,

0 → →XD

−→erX → 0,

(2.14)

with D = D|X

and D = D|X . Just as the sheaf maps D, D have local expres-sions as differential operators, also their restrictions are locally expressible in termsof differential operators: if {f α(zα, θα)} is a section of X, then the correspondingsection {(Df α )(zα, ρα )} of êrX is given by

Df α

zα, ρα

=

∂θ + θ ∂z

f α

zα=zα ,θα=ρα.

Similarly, if {f α (zα, ρα )} is a section of X, then the corresponding section of erX

is Df α

zα , θα

=

∂ρ + ρ∂z

f α

zα=zα ,ρα=θα.

We summarize the relationships between the various sheaves and sheaf maps in thefollowing commutative diagram (of sheaves of -algebras):

0

0

0 / /

/ / X

D / / erX

/ / 0

0 / / X

D

/ / X

D

D / / erX

/ / 0

êrX

êrX

0 0.

(2.15)

We conclude this subsection with the remark that the dualizing sheaf er(X) of the superconformal supercurve (X,X) associated to a supercurve (X,X) is trivial,making (X,X) a super analog of an elliptic curve or a Calabi-Yau manifold; see[DN]. In fact, this statement is true for any N = 2 supercurve (X,) where is anextension of erX by the structure sheaf: if we have

0 → X → →erX → 0,




then has local generators (zα , θα , ρα ) on U α , and on overlaps we get

zβ = F βα

zα, θα

,

θβ = βα zα , θα,

ρβ = βα

zα , θα , ρα

= ber

J(z,θ)

ρα + φβα

zα, θα

,

(2.16)

where ber(J(z,θ)) is the Berezinian of the super Jacobian matrix of the change of (z,θ) coordinates. This is precisely the transition function for erX; see (2.6). Thenthe super Jacobian matrix

J(z,θ,ρ) = ber

∂zF ∂z ∂z

∂θ F ∂θ ∂θ

∂ρ F ∂ρ ∂ρ

= ber

∂zF ∂z ∂z

∂θ F ∂θ ∂θ

0 0 ∂ρ

= ber

J(z,θ)

/∂ρ = 1,

for all overlaps U α ∩ U β , and therefore (X,) has trivial dualizing sheaf.

2.3. Super Riemann surfaces. In this subsection we briefly discuss a special classof N = 1 supercurves, the super Riemann surfaces. This class of curves is studiedwidely in the literature because of its applications in superstring theory; see, forexample, [F], [GN1], [LR], [CR]. (The term SUSY1 curve is used in [Ma1], [Ma2],and superconformal manifold in [RSV].) From our point of view SRSs are specialbecause irreducible divisors and -points can be identified and because there is adifferential operator taking functions to sections of the dualizing sheaf. Both factssimplify the theory considerably. However, by systematically using the duality of theN = 2 superconformal curve, one can extend results previously obtained solely forSRSs to arbitrary supercurves.

In the previous subsection we saw that every N = 1 supercurve (X,X) has a dualcurve (X, X). Of course it can happen that the transition functions of (X,X) are

identical to those of the dual curve (X, X). This occurs if the transition functionssatisfy

DF

zα , θα

=

zα , θα

D

zα , θα

.(2.17)

If (2.17) holds, then the operator Dα = ∂θα + θα ∂zα transforms as

Dβ = (D)−1Dα.(2.18)

So in the situation of (2.17) the supercurve (X,X) carries a (0 | 1)-dimensionaldistribution D such that D2 is nowhere-vanishing (in fact D2 = ∂z). A supercurvecarrying such a distribution is called an (N = 1) SRS. Equivalently, an N = 1 SRS isan (N = 1) supercurve that carries an odd global differential operator with nowhere-

vanishing square that takes values in some invertible sheaf.



SUPERCURVES 11

Recall the Berezinian that occurs in the transformation law for generators of erX,(2.6). It can be written in general as

ber∂zF ∂z

∂θ F ∂θ = DDF

D .

Therefore if (2.17) holds we have ber

∂zF ∂z∂θ F ∂θ

= D so (2.18) tells us that D takes

values in the dualizing sheaf erX.So SRSs are self-dual, as probably first noted in [DRS]. More generally, thequestion

then arises, What happens if the curves (X,X) and (X, X) are isomorphic, but apriori not with identical transition functions? We claim that also in this case thecurve (X,X) is an SRS. Indeed, the operator D restricted to X takes values in thedualizing sheaf êrX of X, as we saw above. Using the isomorphism we can thinkof D as a differential operator taking values in a sheaf isomorphic to the dualizingsheaf erX on X. Since D2

does not vanish we see that (X,X) is an SRS. Nowit is known (and easy to see) that for any SRS there are coordinates such that (2.17)

holds. In these coordinates the transition functions of (X,X) and (X,ˆX) are in factequal.

The N = 2 superconformal curve (X,X) associated to an SRS (X,X) is verysimple. Recall that X is an extension

0 → X →X

−→erX → 0,

where locally (z) = (θ) = 0 and (ρ) = f , with f a local generator of erX. ForSRSs there is a splitting e :erX → X, given locally by e(f) = ρ − θ . One needsto use the definition of an SRS to check that this definition makes global sense, thatis, ρ −θ transforms as a section of erX; for this see [R3]. In other words for an SRSthe associated N = 2 curve has a split structure sheaf:

X = X ⊕erX.

Note that not all SRSs are projected, so there are examples where X is a trivialextension but where (X,X) is not projected.

2.4. Integration on supercurves. Let us first recall the classical situation. On anordinary Riemann surface (X,red

X ) we can integrate a holomorphic 1-form ω alonga contour connecting two points p and q on X. If the contour connecting p and q liesin a single, simply connected, coordinate neighborhood U α with local coordinate zα

we can write ω = df α, with f α ∈ redX (U α) determined up to a constant. The points

p, q are described by the irreducible divisors zα − pα and zα − qα . Then we calculatethe integral of ω along the contour by

q

pω = f α (qα ) − f α (pα ). Suppose next that

p and q are in different coordinate neighborhoods U α and U β , with coordinateszα, zβ related by zβ = F (zα ) on overlaps. Let the contour connecting them lie in




U α ∪ U β and assume that there is on the contour a point r ∈ U α ∩ U β . Then wecan write ω = df α on U α , and ω = df β on U β , with f α (zα) = f β (F(zα )) + cαβ

on overlaps, where cαβ is locally constant on U α ∩ U β . The intermediate point r

can be described by two (corresponding) irreducible divisors zα − rα and zβ − rβ .

Then q

p ω = r

p ω + q

r ω = f β (qβ ) −f β (rβ ) +f α(rα) −f α(pα). This is independentof the intermediate point because the parameter rα in the irreducible divisor zα − rα

transforms as aC-point of the curve: We have rβ = F (rα ) and f α (rα )−f β (rβ ) = cαβ;therefore we can replace r by any other intermediate point in the same connectedcomponent of U α ∩ U β . If p and q are not in adjacent coordinate neighborhoods weneed to introduce more intermediate points.

So there are three crucial facts in the construction of the contour integral of holo-morphic 1-forms on a Riemann surface: the parameter in an irreducible divisor trans-forms as a point, d is an operator that produces from a function on X a section of thedualizing sheaf on X, and the kernel of the operator d consists of the constants. Wefind analogs of all three facts for supercurves.

We have seen that for an N = 1 supercurve in general the parameters in an ir-reducible divisor correspond to a -point of the dual curve. Also the sheaf map D

acting on the dual curve maps sections of X to sections of erX; see (2.15).This suggests that we define a (super-) contour = (γ,P,Q) on (X,X) as

an ordinary contour γ on the underlying topological space X, together with twoirreducible divisors P and Q for (X,X) such that the reduced divisors of P and Q

are the endpoints of γ . So if

P = z − p − θπ , Q = z − q − θχ ,

then the corresponding -points on the dual curve (X, X) are (p, π), (q, χ ); andz = pred, z = qred are the equations for the endpoints of the curve γ . Then we definethe integral of a section {ωα = Df α} of the dualizing sheaf on (X,X) along by

Q

P

ω = Q

P

Df = f q, χ− f p, π.

Here we assume that the contour connecting P and Q lies in a single, simply con-nected, open set. If the contour traverses various open sets, we need to choose inter-mediate divisors on the contour, as before.

A supercontour is called closed if it is of the form = (γ,P ,P ), with theunderlying contour γ closed in the usual sense. Observe that the integral over isindependent of the choice of P , so we omit reference to it.

The contour integration on N = 1 supercurves introduced here seems to be new;it is a nontrivial generalization of the contour integral on SRSs, as described, forinstance, in [F], [M], [Ro]. For closed contours it agrees with the integration theorydescribed in [GKS], [Kh].

We can also understand this integration procedure in terms of the contour integral on

the N = 2 superconformal supercurve (X,X), introduced by Cohn [C]. To this end



SUPERCURVES 13

define on X(U ) ⊕X(U ) the sheaf map (D, D) by the local componentwiseaction of the differential operators Dα

and Dα

as before. Then the square of the

operator (D, D) vanishes and the Poincaré lemma holds for (D, D).

Lemma 2.4.1. Let U be a simply connected, open set on X and let (f,g) ∈

X(U ) ⊕X(U ) such that (D, D)(f,g) = 0. Then there is an element H ∈X(U ) , unique up to an additive constant, such that

(f,g) =

DH, DH

.

Let then(U ) ⊂ X(U )⊕X(U ) be the subsheaf of (D, D)-closed sections.Note that a section of looks in U α like (f α , gα ) = (f(zα, θα),g(zα , ρα )) andfurthermore f is a section of erX and g is a section of êrX. This means that globalizes to the direct sum erX ⊕êrX.

So we get an exact sequence of sheaves:

0 → → X(D,D)

−−−−−−→ → 0.

Now the sections of are the objects on X that can be integrated. A contour forX is a triple (γ ,P ,Q) where P ,Q are two -points of (X,X) having asreduced points the endpoints of the contour γ . Assume that the contour lies in a single,simply connected, open set U . If ω ∈ (U ), then we can write ω = (DH, DH )

for some H ∈ X(U ) and we put Q

P ω = H (Q) − H (P ). The extension tomore complicated contours is as before.

Now start with a section {sα } of erX on (X,X). We can lift it to the section{(sα , 0)} of . In particular, there is a section {H α } of X such that sα = DH α ,DH α = 0. This means that {H α} is in fact a section of the subsheaf X. So inspecifying the -points of (X,X) on the ends of the contour we have the freedomto shift along the fiber of the projection π : (X,X) → (X, X). In other words we

only need to specify -points of the dual curve or, equivalently, irreducible divisorson the original curve.Therefore we can define the integral of a section s = {sα } of erX along a contour

with P , Q two irreducible divisors for (X,X) at the endpoint as follows. We choosetwo -points P and Q of (X,X) that project to the -points of (X, X) cor-responding to P , Q. Then

QP

s = H (Q) − H (P ) if sα = DH and DH = 0.Again we are assuming here that the contour lies in a simply connected region and weextend for the general case using intermediate points. One checks that this procedureof integrating a section of the dualizing sheaf on (X,X) using integration on X

is the same as we had defined before.

2.5. Integration on the universal cover. We consider from now on only holomor-phic (compact, connected, N = 1) supercurves (X,X) of genus g > 1. We fix a pointx0 ∈ X and 1-cycles Ai , Bi , i = 1, . . . , g through x0 with intersection Ai · Bj = δij ,




Ai · Aj = Bi · Bj = 0 as usual. Then the fundamental group π1(X,x0) is generatedby the classes ai , bi corresponding to the loops Ai , Bi , subject solely to the relationa1b1a−1

1 b−11 a2b2a−1

2 b−12 · · · agbg a−1

g b−1g = e.

The universal cover of the supercurve (X,X) is the open superdisk D1|1 =

(D,D1|1 ) of dimension (1 | 1), where D = {z ∈C | |z| < 1} andD1|1 = D ⊗C[θ ],with D the usual sheaf of holomorphic functions on the unit disk. The group G of covering transformations of (D,D1|1 ) → (X,X) is isomorphic to 1(X,x0), andeach covering transformation g is determined by its action on the global coordinates(z,θ) of D1|1. We introduce superholomorphic functions by

F g(z,θ) := g−1 · z, g (z,θ) := g−1 · θ.

If P p is a -point of D1|1, that is, a homomorphism D1|1 → , determined byz → zP ∈ 0, θ → θP ∈ 1, then the action of g in the covering group is defined byg ·P p(f ) = P p(g−1 ·f ). Then zP → F g (zP , θP ) and θP → g(zP , θP ). So -pointstransform as the coordinates under the covering group.

Next consider irreducible divisors P d = z−z1 −θ θ1, Qd = z−z2 −θ θ2. We say thatg · P d = Qd as divisors if we have the identity g−1Qd = P d h(z,θ) as holomorphicfunctions for some invertible h(z,θ). By the same calculation as the one followingLemma 2.2.4, we find that

z2 = F g

z1, θ1

+DF g

z1, θ1

Dg

z1, θ1

g

z1, θ1

,

θ2 =DF g

z1, θ1

Dg

z1, θ1

.

So irreducible divisors transform with the dual action; compare with (2.9).There is a parallel theory for the dual curve: We have a covering (D,D1|1 ) →

(X,ˆX), with covering group

ˆG. The dual covering group

ˆG is isomorphic to G by adistinguished isomorphism: g and g are identified if they give the same transformation

of the reduced disk. Their action on functions is in general different, however, unlesswe are dealing with an SRS. In fact, since duality interchanges irreducible divisorsand -points on the curve and its dual we see that the action of g on the coordinatesis dual to the transformation of g:

g−1 · z = F g (z,θ) +DF g(z,θ)

Dg(z,θ)g (z,θ),

g−1 · θ =DF g(z,θ)

Dg(z,θ).

A function f on (X,X) lifts to a function that is invariant under the covering

group G, and similarly f , a function on (X, X), lifts to a function that is invariant



SUPERCURVES 15

under the dual covering group G. An irreducible divisor or a -point on (X,X) liftsto an infinite set of divisors or points, one for each point on the underlying disk abovethe corresponding reduced point of X.

Let as before x0 be a point on X and d 0 a point on the disk lying over x0. Let γ

be a contour for integration on (X,

X), so γ consists of a contour on X and twoirreducible divisors at the endpoints. The contour lifts to a unique contour on thedisk starting at d 0, and the irreducible divisors lift to unique irreducible divisors for(D,D1|1 ) that reduce to d 0 and the endpoint on the disk, respectively. Also we canpull back sections of erX to (D,D1|1 ) and calculate integrals on (X,X) by liftingto (D,D1|1 ). Since D is simply connected this is a great simplification. For instanceany integral over a closed contour is zero.

Similar considerations apply to the N = 2 curve (X,X) and its universal cover-ing space D1|2 and covering group . Of course D1|2 is the N = 2 curve canonicallyassociated to the N = 1 curve D1|1 as in Subsection 2.2, and the lifts of f ∈ X toD1|2 via either (X,X) or D1|1 as intermediate space coincide.

2.6. Sheaf cohomology for supercurves. Our supercurves are in fact families of

curves over the base scheme (•, ), with the Grassmann algebra of nilpotentconstants. This means that for any coherent sheaf the cohomology groups are finitelygenerated -modules, but they are not necessarily free. This means in particular thatstandard classical theorems, like the Riemann-Roch theorem, do not hold in generalin our situation. (See, for instance, [H].)

The basic facts about sheaf cohomology of families of supercurves are completelyparallel to the classical theory (explained, for instance, in [K]). For a coherent, locallyfree sheaf there exist -homomorphisms α : F → G, with F, G free, finite-rank-modules, that calculate the cohomology. More precisely, for every -module M

we have an exact sequence

0 → H 0(X,⊗ M) → F ⊗ M α⊗1M

−−−−→ G ⊗ M → H 1(X,⊗ M) → 0.(2.19)

Recall from Example 2.1.1 that for any sheaf of X-modules we have an associ-ated split sheaf split =⊗ /m. Therefore, if we choose M = /m, the sequence(2.19) calculates the cohomology groups of the split sheaf split. (These cohomologygroups are Z2-graded vector spaces over /m = C.) Without loss of generality onecan choose the homomorphism α : F → G such that α split = α ⊗1/m is identicallyzero. This means that H 0(X,) (respectively, H 1(X,)) is a submodule (respec-tively, a quotient module) of a free -module of rank dim H 0(X, split) (respectively,of rank dim H 1(X, split)).

We are interested in the question, When are the H i (X,) free? The idea is to checkthis by an inductive procedure, starting with the free cohomology of split. We havefor every j = 1, . . . , n −1 the split exact sequence

0 →m

j

/mj +1

→ /mj +1

→ /mj

→ 0.(2.20)




Since mj /mj +1 ⊗ = mj /mj +1 ⊗C

split, /mi ⊗ = /mi, and is flatover , we obtain, by tensoring with and taking cohomology, the exact sequence(j =m

j /mj +1):

(2.21) 0 → j ⊗CH 0X, split → H 0X,/mj +1→ H 0X,/mjqj

−−→ j ⊗CH 1

X, split → H 1

X,/mj +1

→ H 1

X,/mj

→ 0.

If H 0(X,/mj) and H 1(X,/mj) are free /mj -modules, then the moduleH 0(X,/mj +1) is free over /mj +1 if and only if the connecting map qj in(2.21) is zero if and only if H 1(X,/mj +1) is free as /mj +1-module (see [K,Lemma 10.4]). The relation between the connecting homomorphisms qj and thehomomorphism α that calculates cohomology is as follows: If we assume as aboveα split is zero, then q1 = α ⊗ 1/m2 . More generally, if q1 = q2 = ··· = qj −1 = 0,then qj = α ⊗ 1/mj +1 .

More concretely, we can assume that α is a matrix of size rank G×rank F and theqj are quotients of this matrix by mj +1. Then the cohomology of is the kernel and

cokernel of the matrix α, and the cohomology is free if and only if α is identicallyzero.

If now is an invertible sheaf, split obeys a super Riemann-Roch relation, andin the case of free cohomology we get (hi = rank H i )

h0(X,) − h1(X,) = (deg+1 − g | deg+deg+1 − g),(2.22)

where Xsplit = red

X | . We can relate by Serre duality the cohomology groups of and ∗ ⊗erX; see Appendix A. In particular, H 0(X,∗ ⊗erX) is free if andonly if H 1(X,) is.

We summarize the discussion in this subsection in the following theorem.

Theorem 2.6.1. Let be an invertible X-sheaf. Then H 0(X,) (respectively,

H 1(X,)) is a submodule (respectively, a quotient module) of a free -module of rank dim H 0(X, split) (respectively, of rank dim H 1(X, split)). Furthermore

H 0(X,) is a free -module ⇐⇒ H 1(X,) is free,

⇐⇒ H 0

X,∗ ⊗er X

is free,

⇐⇒ H 1

X,∗ ⊗er X

is free,

in which case the rank of H i (X,) is equal to dim H i (X, split).

2.7. Generic SKP curves

Definition 2.7.1. An SKP curve is a supercurve (X,X) such that the split sheaf X

split is of the formX split = redX |,



SUPERCURVES 17

where is an invertible redX -module of degree zero. If = red

X , then (X,X) iscalled a generic SKP curve.

We discuss in Subsection 4.1 a Krichever map that associates, to an invertiblesheaf on a supercurve (X,X) (and additional data), a point W of an infinite superGrassmannian. If this point W belongs to the big cell (defined below), we obtaina solution of the SKP hierarchy. For W to belong to the big cell it is necessarythat (X,X) is an SKP curve. The generic SKP curves enjoy simple cohomologicalproperties.

Theorem 2.7.2. Let (X,X) be a generic SKP curve. Then the cohomology groups

of the sheaves X,er X are free -modules. More precisely:

H 0

X,X

= | 0, H 1

X,X

= g | g−1,

H 0

X,er X

= g−1 | g, H 1

X,er X

= 0 | .

Proof. Since has no global sections, H 0(X,Xsplit) = C | 0 consists of the

constants only. Now by the definition of a curve over (•, ) we have an inclusion

0 → → X, so H

0

(X,X) contains at least the constants . By Theorem 2.6.1 thenH 0(X,X) must be equal to | 0. Again using Theorem 2.6.1 then also H 1(X,X)

and the cohomology of erX will be free, and the rest of the theorem follows fromthe properties of the split sheaves; see Examples 2.1.1, 2.2.2.

Remark 2.7.3. It is not true that all invertible X-sheaves for a generic SKP curvehave free cohomology. For instance, consider a sheaf with split = X

split, but = X. Then, for a covering {U α } of X, the transition functions of have the formgαβ = 1 + f αβ(z,θ), with f αβ(z,θ) = 0 modulo the maximal ideal m of . Let thenI ⊂ be the ideal of elements that annihilate all f αβ. Then we have H 0(X,) = I

and it is in particular not free.

2.8. Riemann bilinear relations. Let us call sections of erX and êrX holo-morphic differentials (on (X,X) and (X, X), respectively). In this subsection weintroduce analogs of the classical bilinear relations for holomorphic differentials.

Theorem 2.8.1. Let (X,X) be a supercurve and let ω , ω be holomorphic differ-

entials on (X,X) , (X, X) , respectively. Let {ai , bi } be a standard symplectic basis

for H 1(X,Z). Theng

i=1

ai

ω

bi

ω =

gi=1

ai

ω

bi

ω.

Note that we think here of closed contours on the underlying topological space X

as closed supercontours on either (X,X) or (X, X).

Proof. The argument is clearest using the N = 2 curve (X,X) and its universalcovering superdisk D1|2; this way only oneuniversal covering group appears instead

of both G and G. Choose any holomorphic differentials ω on X and ω on X, and




lift them to sections (ω, 0) and (0, ω) of on (X,X). Lifting further to D1|2, let be an antiderivative of (0, ω), so that (D, D) = (0, ω). The crucial pointis that (ω, 0) is itself a section of , because D(ω) = 0. This could not havebeen achieved using only differentials from X. As per the standard argument, we

integrate this object around the polygon obtained by cutting open (X,X). To formthis polygon, fix arbitrarily one vertex P (a -point of the N = 2 disk D1|2) and letthe other vertices be a−1

1 P , b−11 a−1

1 P , . . . , ag b−1g a−1

g · · · b1a1b−11 a−1

1 P , where ai , bi

are the generating elements of . The vertices are the endpoints of supercontourswhose reduced contours are the sides of the usual polygon bounding a fundamentalregion for .

These contours project down to any of X,X,(X,X) as closed loops generatingthe homology; integrating a differential lifted from any of these spaces along a sideof our polygon yields the corresponding period. Labeling the sides of the polygonwith generators of as usual, neighborhoods of the sides labeled ai are identifiedwith each other by bi and vice versa. Then we have

0 =

(ω,0) =

gi=1

ai (ω, 0) −

a

i(ω, 0)

+

gi=1

bi

(ω,0) −

b

i

(ω,0)

.

In the first sum, the two integrals are related by the change of variables given by bi ;the differential (ω, 0) is invariant under this covering transformation while changesby the bi period of ω. The second sum is simplified in the same manner, with theresult

gi=1

ai

ω

bi

ω −

ai

ω

bi

ω

= 0.

2.9. The period map and cohomology. The commutative diagram (2.15) gives acommutative diagram in cohomology that partly reads

H 0

X,êrX

per

H 0

X,êrX

q

H 0

X,erX

per / / H 1(X,)

rep

rep / / H 1

X, X

H 0

X,erX

q / / H 1

X,X

.

(2.23)

Let {ai , bi ; i = 1, . . . , g} be a symplectic basis for H 1(X,Z) and let {a∗i , b∗

i ; i =

1, . . . , g} be a dual basis for H 1(X,Z) and also for H 1(X,). We use Serre



SUPERCURVES 19

duality (see Appendix A.2) to identify H 1(X,X) and H 1(X, X) with the dualsof H 0(X,erX) and H 0(X,êrX).

Lemma 2.9.1. The maps per , per , rep , and rep are explicitly given by

per(ω) =

gi=1

ai

ω

a∗i +

gi=1

bi

ω

b∗i ,

per(ω) =

gi=1

ai

ω

a∗

i +

gi=1

bi

ω

b∗

i ,

rep(σ)(ω) =

gi=1

αi

bi

ω

−

gi=1

βi

ai

ω

,

rep(σ)(ω) =

gi=1

αi

bi

ω

−

gi=1

βi

ai

ω

,

where ω ∈ H 0

(X,er X) , ω ∈ H 0

(X,êr X) , and σ =g

i=1 αi a∗i +βi b

∗i ∈ H

1

(X,).

If we introduce a basis {ωα , α = 1, . . . , g − 1 | wj , j = 1, . . . , g} of H 0(X,erX)

we obtain the period matrix associated to per:

=

ai

ωα

ai

wj bi

ωα

bi

wj

,

where i, j run from 1 to g and α runs from 1 to g −1.For the split curve we have H 0(X,

splitX ) =C |Cg−1, and the map

D : H 0

X, splitX

→ H 0

X,ersplit

X

has as an image a (g −1)-dimensional, even subspace of exact differentials. For theseelements the periods vanish, and one finds the reduction mod m of is given by

split =

0 red(a)

0 red(b)

,

where red =

red(a)

red(b)

is the classical periodmatrix of theunderlying curve (X,red

X ).

By classical results we can choose the basis of holomorphic differentials on the re-duced curve so that red(a) = 1g . From this it follows that we can also choose inH 0(X,erX) a basis such that

ai

wj = δij and so that the period matrix takes theform

= 0 1g

Zo Ze .(2.24)




Note that is not uniquely determined by the conditions we have imposed: we are

still allowed to change → =

0 1g

ZoG Ze+Zo

, corresponding to a change of basis

of H 0(X,erX) by an even invertible matrix

G 0 1g

of size g −1 | g × g −1 | g.

Using the same basis we see that rep has matrix bi

ωα −

aiωα

biwj −

ai

wj

= t I =

Zt

o 0Zt

e −I g

, I =

0 −1g

1g 0

.

Again this matrix is not entirely determined by our choices.From the commutativity of the diagram (2.23) we see that the matrix of the map q

is given by

Q = t I =

0 Zt

o

−Zo Zt e − Ze

.(2.25)

In general, the structure sheaf X and dualizing sheaf êrX of the dual curve donot have free cohomology, so that we cannot represent the maps per, rep, and q by

explicit matrices.The nonfreeness of the cohomology of X and êrX is determined by the odd

component Zo of the period matrix; see (2.24). Recall that splitX = red

X | −1 andêrX

split = | (see Example 2.2.3) and hence

H 0

X, splitX

=C |Cg−1, H 1

X,

splitX

=C

g | 0,

H 0

X,êrXsplit = 0 | Cg , H 1

X,êrX

split =Cg−1 |C.

From the diagram (2.15) we extract in cohomology, using that the map H 1(X,erX)

→ H 2(X,) = | 0 is an (odd!) isomorphism and H 0(X,) = ,

0 →H 0X, X

D−→ H 0X,erX per

−−→ H 1(X,) → H 1X, X→ 0,(2.26)

so that the period map has as kernel H 0(X, X) mod constants and as cokernelH 1(X, X). Therefore per is essentially one of the homomorphisms that calculatecohomology introduced in Subsection 2.6. We can even be more explicit: If {ωα |wj } is the (partially) normalized basis of holomorphic differentials as above, thehomomorphism per maps the submodule generated by the wj isomorphically to a free,rank-g summand of H 1(X,). This is irrelevant for the calculation of cohomology,so we can replace the sequence (2.26) by

0 → H 0

X, X

/

D−→ g−1 Zo

−−→ g → H 1

X, X

→ 0,(2.27)

and H 0(X, X) mod constants is the kernel of Zo, whereas the cokernel of Zo isH 1(X, X).



SUPERCURVES 21

Similarly, the cohomology of êrX is calculated by the sequence

0 → H 0(X,êrX)per

−−→ H 1(X,)rep

−−→H 0

X,erX

∗(2.28)

→H 1X,êrX→ → 0.

The image of a holomorphic differential ω in H 1(X,) is then a vector per(ω) =a(ω)

b(ω)

, where a(ω) and b(ω) are the vectors of a and b periods of ω. By exactness

of (2.28) we have rep ◦ per = 0, or, using bases,Zt

o 0Zt

e −I g

a(ω)

b(ω)

= 0.

This means that the vector b(ω) of b periods is (uniquely) determined by the a

periods: b(ω) = Zt ea(ω), and the vector of a periods is constrained by the equation

Zt oa(ω) = 0. The submodule of H 1(X,) generated by the elements b∗

i maps underrep isomorphically to a free rank-(0 | g) summand of H 1(X,erX)∗, so that for the

calculation of cohomology we can simplify (2.28) to

0 → H 0

X,êrX

→ g Zt

o−−→ g−1 → H 1

X,êrX

→ → 0.(2.29)

We summarize the results on the cohomology of the dual curve in the followingtheorem.

Theorem 2.9.2. Let (X,X) be a generic SKP curve with odd period matrix Zo.

Then

H 0

X, X

/ Ker(Zo), H 1

X, X

Coker(Zo).

Furthermore H 0(X,êr X) Ker(Zt o) and Coker(Zt

o) is a submodule of H 1(X,

êr X) such that

H 1

X,êr X

/Coker

Zt

o

.

2.10. X as an extension of er X. We discuss in this subsection, for genericSKP curves, the structure of X as an extension of erX and the relation with freecohomology and the projectedness of the curve (X,X).

From the sequence (2.7) that defines X, we obtain in cohomology

0 → H 0

X,X

→ H 0

X,X

→ H 0

X,erX

q

−→ H 1

X,X

→ H 1

X,X

→ H 1

X,erX

→ 0.

(2.30)

The cohomology of the sheaves X,erX is given by Theorem 2.7.2. By Theorem2.6.1 (or its extension to rank-two sheaves) H 0(X,X) is a submodule of a g+1 |g−1 and H 1(X,X) is a quotient of a g+1 | g−1. We see from this that the

cohomology of X is free if and only if q is the zero map.




To describe the map q in more detail we need to recall some facts about princi-pal parts and extensions (see, e.g., [K]). For any invertible sheaf let at () andrin() denote the sheaves of rational sections and principal parts for and denoteby at () and rin() their -modules of global sections. Then the cohomology

of

is calculated by0 → H 0(X,) →at () →rin() → H 1(X,) → 0.

In particular we can represent a class α ∈ H 1(X,) as a principal part p =

px ,where px ∈at ()/x , for x ∈ X.

If α ∈ H 1(X,) and ω ∈ H 0(X,), for some other invertible sheaf , then we candefine the cup product ω ∪ α by representing α by a principal part p and calculatingthe principal part ωp =

ωx px in rin(⊗); the image of ωp in H 1(X,⊗)

is then by definition ω ∪ α.We want to understand the kernel of the cup product with ω ∈ H 0(X,) in the

case when ω is odd and free (i.e., linearly independent over ). In this case there arefor any invertible sheaf sections that are immediately annihilated by ω; let thereforeAnn(, ω) ⊂ be the subsheaf of such sections. Putting ω = / Ann(, ω), weget, because ω2 = 0, the exact sequence

0 →ωω

−→ Ann(⊗, ω) → Q → 0.(2.31)

Locally, in an open set U α ⊂ X, we have (U α ) = X(U α )lα, (U α ) = X(U α )mα ,and we write ω = ωα(z,θ)mα , with ωα = φα + θαf α. Then f red

α is a regular functionon U α with some divisor of zeros Df =

ni qi . Some of the qi may also be zeros

of (the lowest-order part of) φα, and there is a maximal gα (z,θ) ∈ X(U α )0 (hereX(U α )0 is the module of even sections) such that

ωα(z,θ) = ωα (z,θ)gα (z,θ)

with gredα a regular function with divisor of zeros Dg (on U α) satisfying 0 ≤ Dg ≤ Df .

Then Ann(⊗

,ω)(U α) is generated by ωα(z,θ)

lα⊗

mα and we see that Q isa torsion sheaf: Q is killed by the invertible sheaf generated locally by the eveninvertible rational function gα(z,θ). Let Dω = {(gα(z,θ),U α )} be the correspondingCartier divisor. Then we have an isomorphism

Ann(⊗, ω) →(Dω), ωα (z,θ)lα ⊗ mα → lα ⊗ 1/gα (z,θ).

The sequence (2.31) is equivalent to

0 →→(Dω) →(Dω)|Dω → 0.

Now the cup product with ω gives a map H 1(ω) → H 1(X,Ann(⊗,ω)) withkernel the image of the natural map φ : H 0(X,Q) → H 1(ω). Identifying H 0(X,Q)

with H 0(X,(Dω)|Dω ), we see that φ is the composition

H 0

X,(Dω)|Dω

→rin() → H 1(X,).



SUPERCURVES 23

Therefore the kernel of ω∪ consists of those α ∈ H 1(X,) that have a representa-tive p ∈ rin() such that ωp has zero principal part, that is, the poles in p arecompensated by the zeros in ω.

Extensions of the form (2.7) are classified by δ ∈ H 1(X,er∗X): We think of X

as a subsheaf of at

(

X⊕

erX) consisting on an open set U of pairs (f,ω) whereω ∈ erX(U ) and f is a rational function such that the principal part f is equal toωp, for p ∈ rin(er∗

X). Then δ ∈ H 1(X,er∗X) is the class of p. It is then easy to

see that the connecting map q : H 0(X,erX) → H 1(X,X) is cup product by theextension class δ: q(ω) = ω ∪ δ. The class δ is represented by the ˇ Cech cocycle

φβα =∂θ F βα

∂θ βα

∈er∗X

U α ∩ U β

,(2.32)

from (2.3), (2.5).

Lemma 2.10.1. Let (X,X) be a generic SKP curve. Then the connecting homo-

morphism q : H 0(X,er X) → H 1(X,X) in (2.30) is the zero map if and only if the

extension (2.7) is trivial. In particular the cohomology of X is free if and only if

the extension is trivial.Proof. It is clear that if the extension is trivial the connecting map q is trivial.From the explicit form, in particular the θ independence, of the cocycle, we see

that it is not immediately killed by multiplication by an odd, free section ω of erX;that is, the cocycle is not zero in the cohomology group H 1(X,er∗

X)/ Ann(er∗X, ω)

if it is nonzero in H 1(X,er∗X). The split sheaf ersplit

X is −1 | . An odd, freesection ω of erX therefore has an associated divisor Dω as constructed above withreduced support included in the divisor of a section of on the underlying curve.Now q(ω) = ω ∪δ is zero if the zeros of ω cancel the poles occurring in the principalpart p representing δ. But by classical results the complete linear system of hasno base points, that is, there is no point on X where where all global sections of vanish. This means that wherever the poles of p occur, there is a section ω of erX

that does not vanish there. So q being zero on all odd generators of H 0(X,erX)implies that the extension is trivial. A fortiori if q is the zero map the extension isalso trivial.

The extension given by the cocycle (2.32) is trivial if

φβα

zα

= σ β

zβ , θβ

− H βα

zα , θα

σ α

zα , θα

(2.33)

for some 1-cochain σ α ∈ er∗X(U α ). In that case, a splitting e : erX → X is

obtained by e(f α ) = ρα − σ α (zα , θα ).

Theorem 2.10.2. For a generic SKP curve (X,X) , X is a trivial extension of

er X if and only if (X,X) is projected.

Proof. We have already observed (in Subsection 2.2) that X projected impliesφβα = 0 in a projected atlas, making the extension trivial.




Now suppose, if possible, that the extension is trivial but that X is not projected.Write the transition functions of X in the form

zβ = f βα

zα

+ θα ηβα

zα

, θβ = ψβα

zα

+ θαBβα

zα

and assume that the atlas has been chosen so that ηβα vanishes to the highest possible(odd) order n in nilpotents. That is, ηβα = 0 mod mn, but not mod mn+2. Writingalso σ α (zα , θα ) = χα (zα)+θα hα (zα ) and substituting in (2.33) yields two conditions.From the θα-independence of φβα one finds that hα mod mn+1 is a global section of −12. Since X is a generic SKP curve, there are no such sections and hα = 0 tothis order. Using this, the second condition becomes

ηβα = Bβα χβ

f βα

− f βα χα modmn+2.

This condition implies that the coordinate change zα = zα − θαχα makes ηβα vanishto an order higher than n, a contradiction.

To the lowest order in nilpotents, the cocycle conditions for the transition functionsof X imply that ηβα /Bβα is a cocycle for H 1(X,−1), while ψβα is a cocycle for

H

1

(X,

−1

). This implies that the projected X’s have codimension (0 | 3g − 3) inthe moduli space of generic SKP curves, which has dimension (4g − 3 | 4g −4) (see[V]). The proof of Theorem 2.10.2 generalizes to a higher order in nilpotents the factthat at the lowest order φβα is a cocycle in H 1(X,−1 |2−1).

2.11. X as an extension of êr X and symmetric period matrices. One canequally view X as an extension of êrX by X. Obviously, if (X, X) is projectedthis extension is trivial, but the converse no longer holds. In the proof of Theorem2.10.2 there is now the possibility that hα = 0. (Recall from Subsection 2.3 that for X

an SRS, a splitting of the extension was universally given by χα = 0, hα = −1.) Onecan see that this extension is not always trivial, however, by constructing exampleswith ψβα = 0 and φβα a nontrivial class. (We are now referring to an atlas for(X, X).) In this subsection we exhibit a connection between the structure of X as

an extension of êrX and the symmetry of the component Ze of the period matrix;see (2.24).

By classical results Zrede is symmetric. However, there seems to be no reason that

Ze is symmetric in general.

Theorem 2.11.1. Let (X,X) be a generic SKP curve and Ze, Zo its (partially)

normalized period matrices (as in (2.24)). Then we have Ze symmetric and Zo = 0if and only if (X,X) is projected.

Proof. This follows immediately from Theorem 2.10.2, Lemma 2.10.1, and theexplicit form (2.25) of the connecting homomorphism q.

Recall the exact sequence

0 → → X (D,ˆ

D)−−−−−−→ → 0,



SUPERCURVES 25

where =erX ⊕êrX is the sheaf of objects that can be integrated on X. Thecorresponding cohomology sequence is in part

0 → → H 0X,X(D,D)

−−−−−−→ H 0(X,)cper

−−−→ H 1(X,)

where cper(ω, ω) = {σ →

σ [ω+ ω]}. Sowe see thatwe can identify H 0(X,X)/

with pairs (ω, ω) of differentials with opposite periods.Now let (ω, ω) be such a pair. ω can bewritten in terms of the basis of H 0(X,erX)

in the form

ω =

ai (ω)wi +

Aαωα,

where ai (ω) denote the a periods and Aα are other constants uniquely determined byω. Then the vector of b periods of ω is

b(ω) = Zea(ω) + ZoA.

Since these coincide with minus the b periods of ω, which are b(ω) = Zt ea(ω) =−Zt

ea(ω), we obtain for each such pair of differentials a relationZe − Zt

e

a(ω) + ZoA = 0.(2.34)

We have a sequence analogous to (2.30) for X as an extension of êrX and aconnecting map q for this situation.

Theorem 2.11.2. Let (X,X) be a generic SKP curve and Ze, Zo its normalized

period matrices. If Ze is symmetric, then q is the zero map.

Proof. Assuming that Ze = Zt e, we determine the set of pairs (ω, ω) with opposite

periods. The a periods of ω can be chosen freely from the kernel of Zt o. According

to (2.34), any ω chosen to match these a periods also has matching b periods if andonly if Aα belongs to the kernel of Zo. Therefore, H 0(X,X) mod constants canbe identified with KerZo ⊕KerZt

o, which is precisely H 0(X, X)/⊕H 0(X,êrX).In this case q is the zero map.

In general it seems that q = 0 does not imply that the extension X of êrX

is trivial, as in Lemma 2.10.1 for the extension of erX by X. Also it seems thatZe = Zt

e cannot be deduced from (2.34) as long as the a periods are constrained tothe kernel of Zt

o.

2.12. Moduli of invertible sheaves. In this subsection we discuss some facts aboutinvertible sheaves on supercurves and their moduli spaces; see also [RSV], [GN2].

An invertible sheaf on (X,X) is determined by transition functions gαβ on over-laps U α ∩ U β , and so isomorphism classes of invertible sheaves are classified by the

cohomology group H 1(X,×X,ev).




The degree of an invertible sheaf is the degree of the underlying reduced sheaf red, with transition functions gred

αβ . Let Pic0(X) denote the group of degree-zeroinvertible sheaves on (X,X). The exponential sheaf sequence

0 →Z

→ X,ev

exp(2π i×·)

−−−−−−−→

×

X,ev → 0(2.35)reduces mod nilpotents to the usual exponential sequence for red

X , and we see thatPic0(X) = H 1(X,X,ev)/H 1(X,Z).

If (X,X) is a generic SKP curve, H 1(X,X) is a free, rank-(g | g −1) -moduleand the map H 1(X,Z) → H 1(X,X) is the restriction of the map H 1(X,) →H 1(X,X), which is dual to the map per of Lemma 2.9.1. So with respect to asuitable basis, H 1(X,Z) → H 1(X,X) is described by the transpose of the periodmatrix (2.24). This implies that the image of H 1(X,Z) is generated by 2g elementsthat are linearly independent over the real part of (see Appendix B for thedefinition of ). The elements of the quotient Pic0(X) = H 1(X,X,ev)/H 1(X,Z)

are the -points of a supertorus of dimension (g | g −1). Each component of Pic(X)

is then isomorphic as a supermanifold to this supertorus.

In general, however, H 1(X,X) is not free, nor is the image of H 1(X,Z) generatedby 2g independent vectors. It seems an interesting question to understand Pic0(X) inthis generality.

For any supercurve (X,X) we define the Jacobian by

Jac(X) = H 0

X,erX

∗odd/H 1(X,Z),

where elements of H 1(X,Z) actby odd linear functionals on holomorphic differentialsfrom H 0(X,erX) by integration over 1-cycles.

We have, as discussed in Appendix A, a pairing of -modules

H 1

X,X

× H 0

X,erX

→ .(2.36)

As we discuss in more detail in Subsection 2.13, invertible sheaves are also describedby divisor classes. We use this in the following theorem.

Theorem 2.12.1. The pairing (2.36) induces an isomorphism of the identity com-

ponent Pic0(X) with the Jacobian Jac(X) given by the usual Abel map: a bundle

∈ Pic0(X) with divisor P − Q corresponds to the class of linear functionals P

Q ,

modulo the action of H 1(X,Z) by addition of cycles to the path from Q to P .

Proof. Let ∈ Pic0(X) have divisor P − Q, with the reduced points P red andQred contained in a single chart U 0 of a good cover of X. If P = z − p − θ π andQ = z − q − θξ , this bundle has a canonical section equal to unity in every otherchart, and equal to

z − p − θ π

z − q − θ ξ =

z − p

z − q −

θ π

z − q + θ ξ

z − p

(z − q)2



SUPERCURVES 27

in U 0. In the covering space H 1(X,X,ev) of Pic0(X), with covering group H 1(X,Z), lifts to a discrete set of cocycles given by the logarithms of the transition functionsof in the chart overlaps, namely,

a0i =

1

2π i

log(z − p) − log(z − q) −

θ π

z − p +

θξ

z − q

in U 0 ∩U i , and zero in other overlaps. The covering group acts by changing the choiceof branches for the logarithms. We now fix the particular choice for which the branch-cut C from Q to P lies entirely in U 0 and meets no other U i . Under the Dolbeaultisomorphism, this cocycle corresponds to a (0, 1)-form most conveniently representedby the current ∂ai in U i , where aij = ai − aj and ∂ = d z∂z + d θ∂θ . It is supported onthe branch cut C, and we can take ai = 0 for i = 0. The pairing (2.36) now associatesto this the linear functional on H 0(X,erX), which sends ω ∈ H 0(X,erX), writtenas f(z) + θφ(z) in U 0, to

X i∂z∂a0ωθ dz d z d θ dθ = X ∂za0ω d z d z dθ

(see [HW]). By the definition of the derivative of a current (refer to [GH]) and Stokestheorem, this can be rewritten

−

∂(X −C)

dz

dθ a0(f + θφ)

= −1

2π i

∂(X −C)

dz

[log(z − p) − log(z − q)]φ +

ξ

z − q−

π

z − p

f

,

where ∂(X − C) denotes the limit of a small contour enclosing C. Using the residuetheorem and the discontinuity of the logarithms across the cut, this evaluates to

C φ dz + πf(p) − ξf(q) =

P

Q ω.

By linearity of the pairing (2.36), we can extend this correspondence to arbitrarybundles of degree zero by taking sums of divisors of the form P i − Qi . In particular,the divisor (P −Q)+(P 1 −P )+(P 2 −P 1)+···+(P n −P n−1)+(P −P n) is equivalentto P − Q, but if the contour P P 1P 2 · · · P nP represents a nontrivial homology class,then the corresponding linear functionals

P

Qdiffer by addition of this cycle to the

integration contour. This shows that the action of H 1(X,Z) specified in the definitionof Jac(X) is the correct one.

2.13. Effective divisors and Poincaré sheaf for generic SKP curves. Another de-scription of invertible sheaves is given by divisor classes.

Recall that a divisor D ∈ Div(X) is a global section of the sheaf at ×ev(X)/×X,ev, so

D, up to equivalence, is given by a collection (f α , U α ) where the f α are even invertible




rational functions that are on overlaps related by an element of ×X,ev(U α ∩ U β ). Each

f α reduces mod nilpotents to a nonzero rational function f redα on the reduced curve,

so that D determines a divisor Dred. Then the degree of D is the usual degree of itsreduction Dred. We havea mappingat ×ev(X) → Div(X), f → (f ), andelements (f )

of the image are called principal. Two even, invertible, rational functions f 1, f 2 giverise to the same divisor if and only if f 1 = kf 2 where k ∈ H 0(X,×X,ev). So if (X,X)

is a generic SKP curve, k is just an even, invertible element of , but in generalmore exotic possibilities for k exist. A divisor D is effective, notation D ≥ 0, if allf α ∈ X,ev(U α ). An invertible X-module can be thought of as a submodule of rank1|0 of the constant sheaf at(X). If (U α) = X(U α )eα, then eα ∈at ×ev(X) and determines the divisor D = {(f α = e−1

α , U α )}. Conversely any divisor D determinesan invertible sheaf X(D) (inat(X)) with local generators eα = f −1

α . Two divisors

D1 = {(f (1)

α , U α )} and D2 = {(f (2)

α , U α )} give rise to equivalent invertible sheaves if and only if they are linearly equivalent , that is, D1 = D2 +(f ) for some element f of at ×ev(X), or more explicitly if and only if f

(1)α = ff

(2)α for all α. If f ∈at ×ev(X) is

a global section of an invertible sheaf = X(D), then D + (f ) ≥ 0 and vice versa.

The complete linear system |D| = |X(D)| of a divisor (or of the correspondinginvertible sheaf) is the set of all effective divisors linearly equivalent to D. So we seethat if = X(D) then

|D| H 0(X,)×ev/H 0(X,X)×

ev.

In the case when the cohomology of is free of rank p + 1 | q and H 0(X,X) is just the constants | 0, the complete linear system |D| is (the set of -points of) asuper-projective space Pp|q

. In particular, if (X,X) is a generic SKP curve and thedegree d of is ≥ 2g −1, the first cohomology of vanishes, the zeroth cohomologyis free of rank d +1 − g | d +1 − g, and |D| P

d −g|d +1−g .

Let X = (X, X) be the dual curve and denote by X(d) the d -fold symmetric productof X; see [DHS]. This smooth supermanifold of dimension (d | d) parametrizeseffective divisors of degree d on (X,X). We have a map (called the Abelian sum)A : X(d) → Picd (X) sending an effective divisor D to the corresponding invertiblesheaf X(D). An invertible sheaf is in the image of A if and only if has an eveninvertible global section: If D ∈ X(d) and = A(D), then the fiber of A at is thecomplete linear system |D|. If the degree d of is at least 2g −1, H 1(X,) is zeroand hence the cohomology of is free. So in that case A is surjective and the fibersof A are all projective spaces Pd −g|d +1−g , and A is in fact a fibration.

The symmetric product X(d) is a universal parameter space for effective divisorsof degree d . This is studied in detail by Domínguez Pérez, Hernández Ruipérez,and Sancho de Salas [DHS]; we summarize some of their results and refer to theirpaper for more details. (In fact they consider curves over a field, but the theoryis not significantly different for curves over .) A family of effective divisors of

degree d on X parametrized by a superscheme S is a pair (S,DS ), where DS is a



SUPERCURVES 29

Cartier divisor on X × S such that for any morphism φ : T → S the induced map(1 × φ)∗X×S (−DS ) → (1 × φ)∗X×S is injective and such that for any s ∈ S therestriction of DS to X × {s} X is an effective divisor of degree d . For example,in X × X(d) there is a canonical divisor (d) such if pD is any -point of X(d)

corresponding to a divisor D, then the restriction of

(d)

to X × {pD} X is justD. Then (X(d) , (d) ) is universal in the sense that for any family (S,DS ) there is aunique morphism : S → X(d) such that DS = ∗(d) .

A family of invertible sheaves of degree d on X parametrized by a superschemeS is a pair (S,S ), where S is an invertible sheaf on X × S such that for anys ∈ S the restriction of S to X × {s} is a sheaf of degree d on X. For example,(X(d) ,

X×X(d) ((d) )) is a family of invertible sheaves of degree d . Two families(S,1), (S,2) are equivalent if 1 = 2 ⊗ π ∗

S , where πS : X × S → S is thecanonical projection and is an invertible sheaf on S . For example, fix a point x of X;then (X(d) ,

X×X(d) ((d) )) is equivalent to (X(d) ,x ), wherex = X×X(d) ((d) )⊗

π ∗X(d)

X×X(d) ((d) )|{x}×X(d)

−1. The family (X(d) ,x ) is normalized: it has the

property that x restricted to {x} × X(d) is canonically trivial. Now consider the

mapping (1×A) : X×X(d) → X ×Picd (X) and the direct image(d)x = (1×A)∗x .

Theorem 2.13.1. Let (X,X) be a generic SKP curve. Let d ≥ 2g −1. Then (d)x

is a Poincaré sheaf on X × Picd (X) , that is, (Picd (X),(d)x ) is a family of invertible

sheaves of degree d that is universal in the sense that for any family (S,) of degree-d

invertible sheaves, there is a unique morphism φ : S → Picd (X) so that = φ∗(d)x .

Furthermore(d)x is normalized so that the restriction to {x}× Picd (X) is canonically

trivial.

2.14. Berezinian bundles. We continue with the study of a generic SKP curve(X,X); we fix an integer n and write for n

X, the Poincaré sheaf on X ×Picn(X).Let s be an invertible sheaf corresponding to s ∈ Picn(X). The cohomology groupsH i (X,s ) vary as s varies over Picn(X) and can in general be nonfree, as we haveseen. Even if the cohomology groups are free -modules, their ranks jump. Still it ispossible to define an invertible sheaf Ber over Picn(X) with fiber at s the line

ber

H 0(X,s )

⊗ ber∗

H 1(X,s )

,

in the case when s has free cohomology. Here ber(M) for a free rank-(d | δ) -module with basis {f 1, . . . , f d , φ1, . . . , φδ } is the rank-1 -module with generatorB[f 1, . . . , f d , φ1, . . . , φδ]. If we are given another basis {f 1, . . . , f d , φ

1, . . . , φδ} = g ·

{f 1, . . . , f d , φ1, . . . , φδ }, with g ∈ Gl(d | δ,), we have the relation

B

f 1, . . . , f d , φ1, . . . , φ

δ

= ber(g)B

f 1, . . . , f d , φ1, . . . , φδ

.

Similarly ber*(M) is defined using the inverse homomorphism ber*. Here ber and

ber* are the group homomorphisms defined in (C.3).




The invertible sheaf s is obtained from the Poincaré sheaf via i∗s. We can re-

formulate this somewhat differently: is an Picn(X)-module and for every -points of Picn(X), via the homomorphism s : Picn(X) → , also becomes an Picn(X)-module, denoted by s . Then s = i∗

s=⊗Picn(X)s . It was Grothendieck’s idea

to study the cohomology of ⊗ M for arbitrary Picn

(X)-modules M . We refer toKempf [K] for an excellent discussion and more details on these matters.The basic fact is that, given the Poincaré bundle on X × Picn(X), there is a

homomorphism α : → of locally free coherent sheaves on Picn(X) such that weget for any sheaf of Picn(X)-modules M an exact sequence

0 → H 0

X ×Picn(X),⊗ M

→⊗ M α×1M

−−−−→ ⊗ M

→ H 1

X ×Picn(X),⊗ M

→ 0.

The proof of this is the same as for the analogous statement in the classical case; see[K].

Now and are locally free, so for small enough open sets U on Picn(X) one

can define ber((U)) and ber∗((U)). This globalizes to invertible sheaves ber()and ber∗(). Next we form the “Berezinian of the cohomology of ” by definingBer = ber() ⊗ ber∗(). Finally one proves, as in Soulé [So, VI.2, Lemma 1], thatBer does not depend, up to isomorphism, on the choice of homomorphism α : → .

Theorem 2.14.1. The first Chern class of the Ber bundle is zero.

We prove this theorem in Subsection 4.2, after the introduction of the infinite superGrassmannian and the Krichever map. The topological triviality of the Ber bundle isa fundamental difference from the situation of classical curves: there the determinantbundle on Pic is ample.

Next we consider the special case of n = g −1. In this case, because of Riemann-Roch (2.22), and have the same rank. Indeed, locally α(U) : (U ) → (U ) is

given, after choosing bases, by a matrix over Picn

(X)(U ) of size d | δ × e | , say. If we fix a -point s in U we get a homomorphism α(U)s : (U ) ⊗ s → (U ) ⊗ s

represented by a matrix over . The kernel and cokernel are the cohomology groupsof s , and these have the same rank by Riemann-Roch. On the other hand if thekernel and cokernel of a matrix over are free we have that the rank of kernel minusthe rank of the cokernel equals d − e | δ − = 0 | 0. So α(U) is a square matrix. Thisallows us to define a map

ber(α) : ber() → ber().

But this is a meromorphic section of ber∗() ⊗ber(), that is, of the dual Berezinianbundle ∗ on Picg−1, because of the nonpolynomial (rational) character of theBerezinian. This section ber(α) is essential for the definition of the τ -function in

Subsection 3.4.



SUPERCURVES 31

2.15. Bundles on the Jacobian; θ -functions. We continue with X being a genericSKP curve. Super θ-functions are defined as holomorphic sections of certain amplebundles on J = Jac(X), when such bundles exist. (As usual, the existence of ampleinvertible sheaves is necessary and sufficient for projective embeddability.) Given one

such bundle, all others with the same Chern class c1 are obtained by tensor productwith bundles having trivial Chern class, so we begin by determining these, that is,computing Pic0(J ). As we briefly discussed in Subsection 2.12 J is the quotient of the affine superspace V =A

g|g−1 = Spec[z1, . . . , zg, η1, . . . , ηg−1] by the lattice L

generated by the columns of the transposed period matrix:

λi : zj → zj + δij , ηα → ηα,

λi+g : zj → zj + (Ze)ij , ηα → ηα + (Zo)iα , i = 1, 2,. . . ,g .(2.37)

We often omit the parity labels e, o on Z, since the index structure makes clear whichis meant.

Any line bundle on such a supertorus J lifts to a trivial bundle on the cover-ing space V . A section of lifts to a function on which the translations λi act by

multiplication by certain invertible holomorphic functions, the multipliers of . Wecan factor the quotient map V → J through the cylinder V /L0, where L0 is thesubgroup of L generated by the first g λi ’s only. Since holomorphic line bundles ona cylinder are trivial, this means that the multipliers for L0 can always be taken asunity. We have Pic0(J ) ∼= H 1(J,ev)/H 1(J,Z). It is very convenient to compute thenumerator as the group cohomology H 1(L,ev) of L acting on the even functionson the covering space V , in part because the cocycles for this complex are precisely(the logarithms of) the multipliers. For the basics of group cohomology, see, for ex-ample, [Si], [Mum]. In particular, factoring out the subgroup L0 reduces our problemto computing H 1(L/L0,L0 ), the cohomology of the quotient group acting on theL0-invariant functions.

A 1-cochain for this complex assigns to each generator of L/L0 an even function(log of the multiplier) invariant under each shift z

j→ z

j+1,

λi+g → F i (z,η) =

n

F in(η)e2π i n·z.

It is a cocycle if the multiplier induced for every sum λi+g + λj +g is independent of the order of addition, which amounts to the symmetry of the matrix i F j giving thechange in F j under the action of λi+g:

F i

zk + Zj k , ηα + Zj α

− F i

zk , ηα

= F j

zk + Zik , ηα + Ziα

− F j

zk , ηα

or, in terms of Fourier coefficients,

F i

n

ηα + Zj α

e2π i

knk Zjk

− F i

n(η) = F j

n

ηα + Ziα

e2π i

knk Zik

− F j

n (η).




One does not have to allow for an integer ambiguity in the logarithms of the multipliersin these equations, precisely because we are considering bundles with vanishing Chernclass. The coboundaries are of the form

λi+g → A(z,η)− Azk + Zik , ηα + Ziαfor a single function A, that is, those cocycles for which

F in(η) = An(η) − An

ηα + Ziα

e2π i

k nk Zik .

This equation has the form

F in(η) = An(η)1 − e2π i

k nk Zik

+ O(Zo).

The point now is that, by the linear independence of the columns of Zrede , for any

n = 0 there is some choice of i for which the reduced part of the exponential in thelast equation differs from unity. This ensures that, for this i, the equation is solvablefor An, first to zeroth order in Zo and then to all orders by iteration. Adding this

coboundary to the cocycle produces one for which F i

n = 0, whereupon the cocycleconditions imply F

j

n = 0 for all n = 0 and all j as well.Thus any nontrivial cocycles are independent of zi . In the simplest case, when

the odd period matrix Zo = 0, all such cocycles are indeed nontrivial, and we havean analog of the classical fact that bundles of trivial Chern class are specified by g

constant multipliers. Here a cocycle is specified by giving g even elements F i0(η) in

the exterior algebra [ηα] (elements of H 0(J,J )), leading to dim Pic0(J ) = g2g−2|

g2g−2(the number of ηα is g −1). In general, when Zo = 0, not all cochains specified

in this way are cocycles, and some cocycles are trivial: Pic0(J ) is smaller and ingeneral is not a supermanifold.

As to the existence of ample line bundles, let us examine in the super case theclassical arguments leading to the necessary and sufficient Riemann conditions (see

[GH], [LB]). The Chern class of a very ample bundle is represented in de Rhamcohomology by a (1, 1)-form obtained as the pullback of the Chern class of the hy-perplane bundle via a projective embedding. We can introduce real, even coordinatesxi , i = 1, . . . , 2g for J dual to the basis λi of the lattice L, meaning that xj → xj +δij

under the action of λi . The associated real, odd coordinates ξα, α = 1, . . . , 2g −2 canbe taken to be globally defined because every real supermanifold is split. The relationbetween the real and complex coordinates can be taken to be

zj = xj +

gi=1

Zij xi+g , j = 1,. . . ,g ,

ηα = ξα + iξα+g−1 +

g

i=1

Ziα xi+g , α = 1, . . . , g −1.



SUPERCURVES 33

The de Rham cohomology is isomorphic to that of the reduced torus and can berepresented by translation-invariant forms in the dxi . The Chern class represented bya form

g

i=1 δi dxi dxi+g is called a polarization of type = diag(δ1, . . . , δg) withelementary divisors the positive integers δi . We consider principal polarizations δi = 1

only, because nontrivial nonprincipal polarizations generically do not exist, even onthe reduced torus [L]. Furthermore, a nonprincipal polarization is always obtained bypullback of a principal one from another supertorus whose lattice L contains L as asublattice of finite index [GH]. Reexpressing the Chern form in complex coordinates,the standard calculations lead to the usual Riemann condition Ze = Zt

e to obtain a(1, 1)-form. Together with the positivity of the imaginary part of the reduced matrix,the symmetry of Ze (in some basis) is necessary and sufficient for the existence of a (1, 1)-form with constant coefficients representing the Chern class. This can beviewed as the cocycle condition, symmetry of i F j , for the usual multipliers of aθ-bundle, F j = −2π izj .

The usual argument that the (1, 1)-form representing the Chern class can always betaken to have constant coefficients depends on Hodge theory, particularly the Hodgedecomposition of cohomology, for a Kähler manifold such as a torus. This does not

hold in general for a supertorus with Zo = 0. For example, H 1dR(J ) is generated bythe 2g 1-forms dxi , whereas H 1,0(J ) contains the g | g −1 nontrivial forms dzi , d ηα ,with certain nilpotent multiples of the latter being trivial. Indeed, since by (2.37) ηα isdefined modulo entries of column α of Zo, ηα is a global function and d ηα is exactwhen ∈ annihilates these entries. Thus, H 1,0(J ) cannot be a direct summandin H 1dR(J ). Correspondingly, some η-dependent multipliers F j = −2π izj + · · · maysatisfy the cocycle condition and give ample line bundles. We do not know a simplenecessary condition for a Jacobian to admit such polarizations.

When Ze is symmetric, we can construct θ-functions explicitly. Consider first thetrivial case with Zo = 0 as well. Then the standard Riemann θ-function (z; Ze)

gives a super θ-function on Jac(X), where (z; Ze) is defined by Taylor expansionin the nilpotent part of Ze as usual. It has of course the usual multipliers,

zj + δij ; Ze

=

zj ; Ze

,

zj + Zij ; Ze

= e−π i(2zi +Zii )

zj ; Ze

.

(2.38)

Multiplication of (z; Ze) by any monomial in the odd coordinates ηα gives another,even or odd, θ-function having the same multipliers, whereas translation of the argu-ment z by polynomials in the ηα leads to the multipliers for another bundle with thesame Chern class.

In the general case with Zo = 0, θ-functions with the standard multipliers can beconstructed as follows. Such functions must obey

H

zj + δij , ηα; Z

= H

zj , ηα ; Z

,

H

zj + Zij , ηα + Ziα ; Z

= e−π i(2zi +Zii )

H

zj , ηα ; Z

.




The function (z; Ze) is a trivial example independent of η; to obtain others onechecks that when H satisfies these relations then so does

H α = ηα +1

2π i

k

Zkα

∂

∂zkH.

Applying this operator repeatedly one constructs super θ-functions α···γ reducingto ηα · · · ηγ (z; Ze) when Zo = 0.

“Translated” θ-functions, which are sections of other bundles having the sameChern class, can be obtained by literally translating the arguments of these only inthe simplest cases. Constant shifts in the multiplier exponents F j can be achievedby constant shifts of the arguments zj . Shifts linear in the ηα are obtained by zj →zj +ηα αj , which is a change in the chosen basis of holomorphic differentials on X;see the discussion after (2.24). The resulting θ-functions have the new period matrixZe +Zo. More generally, translated θ-functions can be obtained by the usual methodof determining their Fourier coefficients from the recursion relations following from

the desired multipliers. We do not know an explicit expression for them in terms of conventional θ-functions.

It is easy to see that any meromorphic function F on the Jacobian can be ra-tionally expressed in terms of the θ-functions we have defined. Expand F(z,η) =

I J βI ηJ F I J (z) in the generators of [ηα ], with multi-indices I, J . Then the zeroth-order term F 00 is a meromorphic function on the reduced Jacobian, hence a rationalexpression in ordinary θ-functions. Using Ze as the period matrix argument of theseθ-functions gives a meromorphic function on the Jacobian itself, whose reductionagrees with F 00. Subtract this expression from F to get a meromorphic function onthe Jacobian whose zeroth-order term vanishes, and continue inductively, first in J ,then in I . For example, F 0α is equal, to the lowest order in the β’s, to a rationalexpression in θ-functions of which one numerator factor is a α . Subtracting this

expression removes the corresponding term in F while only modifying other termsof higher order in the β’s.

3. Super Grassmannian, τ -function, and Baker function

3.1. Super Grassmannians. In this subsection we introducean infinite super Grass-mannian and related constructions. The infinite Grassmannian of Sato [S] or of Segaland Wilson [SW] consists (essentially) of “half-infinite-dimensional” vector sub-spaces W of an infinite-dimensional vector space H such that the projection on afixed subspace H − has finite-dimensional kernel and cokernel. In the super categorywe replace this by the super Grassmannian of free, “half-infinite-rank” -modules of an infinite-rank, free -module H such that the kernel and cokernel of the projectionon H − are a submodule and a quotient module, respectively, of a free, finite-rank -

module. In [Sc1] a similar construction can be found, but it seems that there =C



SUPERCURVES 35

is taken (as is also the case in [Mu]). This is too restrictive for our purposes involvingalgebraic supercurves over nonreduced base ring .

Let ∞|∞ be the free -module [z, z−1, θ ] with z an even and θ an odd variable.We introduce the notation

ei = zi , ei−(1/2) = zi θ, i ∈ Z.(3.1)

We think of an element h =∞

i=−N hi ei , hi ∈ of ∞|∞ not only as a series in z, θ

but also as an infinite column vector:

h = (... , 0, . . . , h−1, h−(1/2), h0, h(1/2), h1, . . . , 0, . . . )t .

We introduce on ∞|∞ an odd Hermitian product

f(z,θ),g(z,θ) =1

2πi

dz

zdθf(z,θ)g(z,θ)(3.2)

=1

2πi f 0g1 + f 1g0dz

z,

where f(z,θ) is the extension of the complex conjugation of (see Appendix B) to∞|∞ by z = z−1 and θ = θ , and where f(z,θ) = f 0 + θf 1, and similarly for g. LetH be the completion of ∞|∞ with respect to the Hermitian inner product.

We have a decomposition H = H − ⊕H +, where H − is the closure of the subspacespanned by ei for i ≤ 0, and H + is the closure of the space spanned by ei with i > 0,for i ∈ (1/2)Z.

The super Grassmanniangr is the collection of all free, closed -modules W ⊂ H

such that the projection π− : W → H − is super Fredholm, that is, the kernel andcokernel are a submodule and a quotient module, respectively, of a free, finite-rank-module.

Example 3.1.1. Let W be the closure of the subspace generated by δ + z, θ andzi , zi θ for i ≤ −1, for δ a nilpotent, even constant. Let A ⊂ be the ideal of annihilators of δ. Then W is free, the kernel of π− is A(δ + z) ⊂ (δ + z), and thecokernel is isomorphic to /δ.

Let I be the subset {i ∈ (1/2)Z | i ≤ 0}. We consider matrices with coefficients in of size (1/2)Z× I :

= (W ij ), where i ∈1

2Z, j ∈ I.

An even matrix of this type is called an admissible frame for W ∈ gr if the closureof the subspace spanned by the columns of is W and if, moreover, in the decom-

position= W −W + induced by H = H − ⊕H +, the operator W − : H − → H − differs

from the identity by an operator of super trace class and W + : H − → H + is compact.




Let Gl(H −) be the group of invertible maps 1+ X : H − → H − with X super traceclass. Then the super frame bundle fr, the collection of all pairs (W,) with anadmissible frame for W ∈ gr, is a principal Gl(H −) bundle over the super Grass-mannian. Elements of Gl(H −) have a well-defined Berezinian; see [Sc1] for some

details. This allows us to define two associated line bundles Ber(

gr) and Ber

*

(

gr)on gr. More explicitly, an element of Ber(gr) is an equivalence class of triples(W,, λ), with a frame for W , λ ∈ ; here (W,g,λ) and (W,, ber(g)λ) areequivalent for g ∈ Gl(H −). For Ber*(gr) replace ber(g) by ber*(g). For simplicitywe write (, λ) for (W,, λ), as determines W uniquely.

The two bundles Ber(gr) and Ber*(gr) each have a canonical section. Let be

a frame for W ∈ gr and write =

W −W +

as above. Then

σ(W) =, ber(W −)

, σ ∗(W ) =

, ber*(W −)

(3.3)

aresections of Ber*(gr),Ber(gr), respectively. It is a regrettable fact of life that nei-ther of these sections is holomorphic; indeed there are no global sections to Ber(gr)

or Ber*

(gr) at all; see [Ma1]. This is a major difference between classical geometryand supergeometry.

3.2. The Chern class of Ber(gr) and the gl∞|∞ cocycle. First we summarizesome facts about complex supermanifolds that are entirely analogous to similar factsfor ordinary complex manifolds. Then we apply this to the super Grassmannian,following the treatment in [PS] of the classical case.

Let M be a complex supermanifold. The Chern class of an invertible sheaf onM is an element c1() ∈ H 2(M,Z). By the sheaf inclusion Z→ and the de Rhamtheorem H 2(M,) H 2dR(M), we can represent c1() by a closed 2-form on M .On the other hand, if ∇ : → ⊗1, with 1 the sheaf of smooth 1-forms, is aconnection compatible with the complex structure, then the curvature F of ∇ is alsoa 2-form. By the usual proof (see, e.g., [GH]) we find that c1() and F are equal, up

to a factor of i/2π .We can locally calculate the curvature of an invertible sheaf by introducing a

Hermitian metric , on it: if s, t ∈ (U ), then s, t (m) is a smooth function inm ∈ U taking values in , linear in t , and satisfying s, t (m) = t, s(m). Choosea local generator e of and let h = e, e. The curvature is then F = ∂∂ log h, with∂ =

dzi (d/dzi ) +

dθα (∂/∂θα) and ∂ defined by a similar formula.

Now consider the invertible sheaf Ber(gr) on gr. If s = (, λ) is a section,the square length is defined to be s, s = λλber(H ), where the superscript H

indicates conjugate transpose. Of course, this metric is not defined everywhere ongr because of the rational character of ber, but we are interested in a neighborhood

of the point W 0 with standard frame 0 =

1H −0

where there is no problem. The

tangent space at W 0 can be identified with the space of maps H − → H + or, more

concretely, by matrices with the columns indexed by I = {i ∈ (1/2)Z | i ≤ 0} and



SUPERCURVES 37

with rows indexed by the complement of I . Let x, y be two tangent vectors at W 0.Then the curvature at W 0 is calculated to be

F(x,y) = ∂∂ log h(x,y) = Str

xH y − yH x

,(3.4)

where we take as local generator e = σ , the section defined by (3.3), so that h =σ, σ . We can map the tangent space at W 0 to the Lie superalgebra gl∞|∞() via

x →

0 −xH

x 0

. Here gl∞|∞() is the Lie superalgebra corresponding to the Lie

supergroup Gl∞|∞() of infinite, even, invertible matrices g (indexed by (1/2)Z)with block decompostion

a bc d

with b, c compact and a, d super Fredholm. We see

that (3.4) is the pullback under this map of the cocycle on gl∞|∞() (see also [KL])given by

c : gl∞|∞() × gl∞|∞() →

(X,Y) →1

4Str(J [J, X][J, Y ]),

(3.5)

where J =

1H − 00 −1H +

. In terms of the block decomposition of X,Y we have

c(X,Y) = Str

cXbY − bXcY

.

The natural action of Gl∞|∞() on gr lifts to a projective action on Ber(gr);the cocycle c describes infinitesimally the obstruction for this projective action to bea real action. Indeed, if g1 = exp(f 1), g2 = exp(f 2), and g3 = g1g2 are all in theopen set of Gl∞|∞() where the −− blocks ai are invertible, the action on a point of Ber(gr) is given by

gi ◦ (, λ) =

gia−1i , λ

.(3.6)

(One checks as in [SW] that if is an admissible basis, then so is gg

−1

−−.) Thenwe haveg1 ◦ g2 ◦ (, λ) = exp[c(f 1, f 2)]g3 ◦ (,λ).

We can also introduce the projective multiplier C(g1, g2) for elements g1 and g2 thatcommute in Gl∞|∞():

g1 ◦ g2 ◦ g−11 ◦ g−1

2 (, λ) = C(g1, g2)(,λ),(3.7)

where C(g1, g2) = exp[S(f 1, f 2)] if gi = exp(f i ) and

S(f 1, f 2) = Str([f 1, f 2]).(3.8)

In Subsection 4.2 we use the projective multiplier to show that the Chern class of the

Berezinian bundle on Pic0(X) is trivial.




3.3. The Jacobian super Heisenberg algebra. In the theory of the KP hierarchy animportant role is played by a certain Abelian subalgebra of the infinite matrix algebraand its universal central extension, loosely referred to as the (principal) Heisenbergsubalgebra. In this subsection we introduce one of the possible analogs of this algebra

in the super case.Let the Jacobian super Heisenberg algebra be the -algebraHeis = [z, z−1, θ ].Of course, this is, as a -module, the same as ∞|∞ but now we allow multiplicationof elements. When convenient we identify the two; in particular, we use the basis{ei } of (3.1) also for Heis. We think of elements of Heis as infinite matrices ingl∞|∞(): if Eij is the elementary matrix with all entries zero except for the ijthentry, which is 1, then

ei =n∈Z

En+i,n + En+i−(1/2),n−(1/2), ei−(1/2) =n∈Z

En+i−(1/2),n.

We have a decomposition Heis = Heis− ⊕ Heis+ in subalgebras Heis− =z−1[z−1, θ ] and Heis+ = [z, θ ]. Elements of Heis+ correspond to lower trian-

gular matrices and elements of Heis− to upper triangular ones. By exponentiation weobtain from Heis− and Heis+ two subgroups G− and G+ of Gl∞|∞(), generatedby

g±(t ) = exp

i∈±I

t i ei

,

where t i ∈ is homogeneous of the same parity as ei (and t i is zero for almost all i,say).

For an element g =

a bc d

of G+ the block b vanishes, whereas if g ∈ G− the block

c = 0. In either case the diagonal block a is invertible and we can lift the action of either G− or G+ to a (potentially projective) action on Ber(gr) and Ber*(gr), via(3.6). Since the cocycle (3.5) is zero when restricted to both Heis− and Heis+ we

get an honest action of the Abelian groups G± on Ber(gr) and Ber*

(gr), just asin the classical case.In contrast with the classical case, however, as was pointed out in [Sc1], the actions

of G− and G+ on the line bundles Ber(gr) and Ber*(gr) mutually commute. Thisfollows from the next lemma.

Lemma 3.3.1. Let g± ∈ G± and write a± = exp(f ±) , with f ± ∈ gl(H −). Then

StrH −

f −, f +

= 0,

so that the actions of g− and g+ on Ber(gr) and Ber*(gr) commute.

Proof. The elements f ± act on H − by multiplication by an element of Heis±,followed by projection on H − if necessary. So write f ± = πH − ◦i>0 c±

i z±i +

γ ±i z±i θ . To find the supertrace we need to calculate the projection on the rank-(1 | 0)



SUPERCURVES 39

and -(0 | 1) submodules of H − generated by z−i and z−i θ :

f +f −z−k |z−k = f +

i>0

c−i z−i−k

z−k

=

i>0

c+i c−

i

z−k ,

f +f −z−kθ |z−k θ =

i>0

c+i c−

i

z−k θ,

f −f +z−k |z−k = f −

ki=1

a+i zi−k

z−k

=

ki=1

c+i c−

i

z−k,

f −f +z−kθ |z−k θ = f −

ki=1

c+i zi−kθ

z−k θ

=

ki=1

c+i c−

i

z−k θ.

Since the supertrace is the difference of the traces of the restrictions to the even andodd submodules, we see that Str([f +, f −]) = 0 so that, by (3.7), (3.8), the actions of G± on Ber(gr) commute.

3.4. Baker functions, the full super Heisenberg algebra, and τ -functions. We de-fine W ∈ gr to be in the big cell if it has an admissible frame (0) of the form

(0) =

. . ....

......

...

. . . 1 0 0 0

. . . 0 1 0 0

. . . 0 0 1 0

. . . 0 0 0 1∗ ∗∗ ∗ ∗ ∗ ∗

,

that is, ((0))− is the identity matrix. Note that the canonical sections σ and σ ∗ donot vanish, or blow up, at a point in the big cell.

If is any frame of a point W in the big cell we can calculate the standard frame(0) through quotients of Berezinians of minors of . Indeed, if we put A = −,then the maximal minor A of is invertible and we have

(0)A =.(3.9)

Write (0) =

w(0)ij Eij . Then we can solve (3.9) by Cramer’s rule, (C.4), to find

for i > 0, j ≤ 0:

w(0)ij =

ber

Aj (ri )

ber(A), if j ∈ Z,

ber* Aj (ri )

ber*(A), if j ∈ Z+ (1/2).

Here Aj (ri ) is the matrix obtained from A by replacing the j th row by ri , the ith row

of . In particular the even and odd “Baker vectors” of W , that is, the zeroth and




−(1/2)th column of (0), are given by

w0 = e0 +i>0

i∈(1/2)Z

ber

A0(ri )

ber(A)ei ,

w1 = e−(1/2) +i>0

i∈(1/2)Z

ber*

A−(1/2)(ri )

ber*(A)ei .

(3.10)

The corresponding “Baker functions” are obtained by using ei = zi , ei−(1/2) = zi θ .Then (3.10) reads

w0(z,θ) = 1 +i>0

ziber

A0(ri )

+ber

A0(ri−(1/2))

θ

ber(A),

w1(z,θ) = θ +i>0

ziber* A−(1/2)(ri )

+ber* A−(1/2)(ri−(1/2))

θ

ber*(A).

(3.11)

Here and henceforth (unless otherwise noted) the summations run over (subsets of)the integers.

The full super Heisenberg algebraHeis is the extension Heis[d/dθ ] = [z, z−1,

θ ][d/dθ ]. This is, just as the Jacobian super Heisenberg algebra, a possible analogof the principal Heisenberg of the infinite matrix algebra used in the standard KPhierarchy; see [KL]. Heis is non-Abelian and the restriction of the cocycle (3.5) toit is nontrivial, in contrast to the subalgebra Heis.Heis acts in the obvious way on ∞|∞ and we can represent it by infinite matrices

from gl∞|∞(). We introduce a basis for Heis by

λ(n) = z−n

1 − θ

d

dθ

=

k∈Z

Ek,k+n, f (n) = z−n d

dθ=

k∈Z

Ek,k+n−(1/2),

µ(n) = z−nθd

dθ=k∈Z

Ek−(1/2),k−(1/2)+n, e(n) = z−nθ =k∈Z

Ek−(1/2),k+n.

We can rewrite the Baker functions as quotients of Berezinians, using Heis. To thisend define the following even, invertible matrices (over the ring [u,φ,∂/∂φ]):

Q0(u,φ) = 1 +

∞n=1

un[λ(n) + f (n)φ],

Q1(u,φ) = 1 +

∞n=1

un

µ(n) + e(n)

∂

∂φ

,

where u (respectively, φ) is an even (respectively, odd) variable. We can let these

matrices act on H and obtain in this way infinite vectors over the ring [u,φ,∂/∂φ].



SUPERCURVES 41

Also we can let these matrices act on an admissible frame and obtain a matrix over[u,φ,∂/∂φ].

Lemma 3.4.1. Let w0(u,φ) and w1(u,φ) be the even and odd Baker functions of

a point W in the big cell. For any frame of W we have

w0(u,φ) =ber

Q0(u,φ)

−

ber(A)

, w1(u,φ) =ber* Q1(u,φ)

−

φ

ber*(A),

with A =−.

Proof. Let ri , ri,0, and ri,1 be, respectively, the ith row of , Q0(u,φ), andQ1(u,φ). Then one calculates that for i ∈ Z we have ri−(1/2),0 = ri−(1/2) andri,1 = ri and also

ri,0 = ri +k≥1

uk

ri+k + ri+k−(1/2)φ

= ri + u(ri+1,0 + ri+(1/2),0φ),

ri−(1/2),1 = ri−(1/2) +k≥1

uk

ri+k−(1/2) + ri+k

∂

∂φ

= ri−(1/2) + u

ri+1−(1/2),1 + ri+1,1

∂

∂φ

.

(3.12)

Let X be an even matrix. Because of the multiplicative property of Berezinians wecan add multiples of a row to another row of X without changing ber(X) and ber*(X).Using such row operations we see, using (3.12), that

ber[Q0(u,φ)]−

= ber

A0(r0,0)

,

ber* [Q1(u,φ)]−= ber* A−(1/2)r−(1/2),1.

Now ber is linear in even rows, and ber * in odd rows, so by (3.12) we find

ber

Q0(u,φ)

−

= ber(A)+

i>0

ui

ber

A0(ri )

+ber

A0

ri−(1/2)

φ

,

and

ber* [Q1(u,φ)]−

= ber*(A)

+i>0

ui

ber* A−(1/2)

ri−(1/2)

+ber* A−(1/2)(ri )

∂

∂φ

.

Comparing with (3.11) proves the lemma.




We now consider the flow on gr generated by the negative part of the JacobianHeisenberg algebra. Define

γ(t) = expi>0

t i z−i + t i−(1/2)z−i θ, t i ∈ ev, t i−(1/2) ∈ odd(3.13)

and put for W ∈ grW(t) = γ(t)−1W.

The τ -functions associated to a point W in the big cell are then functions onHeis−,ev:

τ W (t),τ ∗W (t ) : Heis−,ev → ∪{∞}(3.14)

given by

τ W (t) =σ

γ(t)−1W

γ(t)−1σ(W)=

ber

[γ(t)−1 ◦]−

ber([]−)

,(3.15)

τ ∗W (t) =

σ ∗γ(t)−1W γ(t)−1σ ∗(W ) =

ber* [γ(t)−1 ◦]−ber*

[]−

.(3.16)

Here σ and σ ∗ are the sections of Ber*(gr) and Ber(gr) defined in (3.3) andγ −1 ∈ Gl∞|∞() acts via (3.6) on Ber(gr) and Ber*(gr).

The Baker function of W becomes now a function on Heis−,ev, and we have anexpression in terms of a quotient of (shifted) τ -functions:

w0(t ; u,φ) =τ W

t ; Q0

τ W (t )

, w1(t ; u,φ) =τ ∗W

t ; Q1

τ ∗W (t )

,(3.17)

where

τ W

t ; Q0

=

ber [Q0]−ber(−) , τ

∗W (t ; Q1) =

ber* [Q1]−φ

ber*(−) .

Note that even if we are only interested in the Jacobian Heisenberg flows, the fullHeisenberg flows automatically appear in the theory if we express the Baker functionsin terms of the τ -functions. In principle we could also consider the flows on grgenerated by the full super Heisenberg algebra Heis. However, since Heis is non-Abelian the interpretation of these flows is less clear and therefore we leave thediscussion of these matters to another occasion.

4. The Krichever map and algebro-geometric solutions

4.1. The Krichever map. Consider now a set of geometric data (X,P,(z,θ),, t ),where

• X is a generic SKP curve as before;



SUPERCURVES 43

• P is an irreducible divisor on X, so that P red is a single point of the underlyingRiemann surface Xred;

• (z,θ) are local coordinates on X near P , so that P is defined by the equationz = 0;

• is an invertible sheaf on X;• t is a trivialization of in a neighborhood of P , say, U P = {|zred| < 1}.

We associate to this data a point of the super Grassmannian gr.For studying meromorphic sections of we have the exact sequence

0 →inc

−−→(P )res

−−→P red ∼= | → 0,(4.1)

which gives

H 0() → H 0(P )

→ | → H 1() → H 1

(P )

→ 0,(4.2)

where the residue is the pair of coefficients of z−1 and θ z−1 in the Laurent expansion.Let(∗P ) = limn→∞(nP) be the sheaf of sections of holomorphic except pos-

sibly for a pole of arbitrary order at P . The Krichever map associates to a set of geo-metric data as above the -module of formal Laurent series W = zt [H 0(X,(∗P ))],which is viewed as a submodule of H .

In [MuR], [R1] the concern was expressed that W might not be freely generatedand, hence, not be an element of gr as we have defined it. However, we have thefollowing.

Theorem 4.1.1. H 0(X,(∗P )) is a freely generated -module, and W ∈ gr.

Further, W belongs to the big cell if the geometric data satisfy H 0(X,) = H 1(X,)

= 0 , which happens generically if deg= g −1.

Proof. Assume first that H 0(X,) = H 1(X,) = 0. Then the sequence (4.2)applied to gives

0 → H 0(P )

→ | → 0 → H 1

(P )

→ 0,(4.3)

so that H 0(X,(P)) is freely generated by an even and odd section having principalparts z−1 and θ z−1, and H 1(X,(P)) is still zero. Applying the same sequenceinductively to (nP) shows that H 0(X,(∗P )) is freely generated by one evenand one odd section of each positive pole order. So W is obtainable from H − bymultiplication by a lower triangular invertible matrix, and W belongs to the bigcell of gr. We also have H i ( split) = 0, i = 0, 1, from Theorem 2.6.1. And, bythe super Riemann-Roch theorem (2.22), deg = deg split = g − 1. Moreover, bysemicontinuity, in Picg−1(X) the cohomology groups H i () can only get larger onZariski-closed subsets, so generically they are zero.

Now consider the general situation in which H i () may not be zero. Still, by

twisting, H 1((nP)) = H 1( split(nP)) = 0 for n sufficiently large. Then, by the




previous argument, H 0((∗P )) has non–purely nilpotent elements with poles of order n + 1 and higher; the worry is that one may only be able to find nilpotentgenerators for H 0((nP)). So take f ∈ H 0((nP)) of order k in nilpotents: itsimage in H 0((nP)/mk) is zero, but its image f in H 0((nP)/mk+1) is nonzero

and also lies in

k

H

0

(

split

(nP)). Thenˆ

f can be identified with a sum of elementsf a of H 0( split(nP)) with coefficients from k. By the extension sequence (2.21),each f a can be extended order-by-order in nilpotents to an element of H 0((nP))

that is not purely nilpotent. So we can write the order-k element f as a -linearcombination of non–purely nilpotent elements of H 0((nP)), modulo an element of order k +1. Induction on k shows then that any element of H 0((nP)) is a -linearcombination of non–purely nilpotent elements of H 0((nP)). So there exists a set of nonnilpotent elements that span H 0((nP)) over . A linearly independent subsetof these completes a basis for H 0((∗P )).

4.2. The Chern class of the Ber bundle on Pic0(X). By the arguments of theprevious subsection we have, in the case when W ∈ gr is obtained by the Krichevermap from geometric data (X,P,,(z,θ),t), an exact sequence of -modules:

0 → H 0(X,) → W → H − → H 1(X,) → 0.(4.4)

We can interpret the Ber bundle Ber(gr) in terms of this sequence as follows. LetM be a free -module, possibly of infinite rank, and let B = {µ} be a collection of bases for M such that any two bases µ, µ ∈ B are related by µ = µT where T ∈Aut(M) has a well-defined Berezinian. Then we associate to the pair (M,B) a free,rank-(1 | 0) module ber(M) with generator b(µ) for any µ ∈ B with identificationb(µ) = ber(T )b(µ).

The fiber of Ber(gr) at W can then be interpreted as ber(W ), using the collectionof admissible bases as B in the above definition. Similarly we can construct on gr aline bundle with fiber at W the module ber(H −). Clearly this bundle is trivial, so wecan, even better, think of Ber(gr) as having fiber ber(W ) ⊗ ber*(H

−). But by the

properties of the Berezinian we get from (4.4)

ber(W ) ⊗ber*(H −) = ber

H 0(X,)

⊗ber* H 1(X,)

.

Now we saw in Subsection 2.14 that the Ber bundle Ber (Pic0(X)) on Pic0(X) hasthe same fiber, with the difference that there we were dealing with bundles of degreezero and here has degree g −1.

For fixed (X,P,(z,θ)) the collection M consisting of Krichever data (X,P,(z,θ),

, t ) forms a supermanifold and we have two morphisms

i : M →gr, p : M → Pic0(X)

where i is the Krichever map and p is the projection from Krichever data to the line



SUPERCURVES 45

bundle. (Here we identify Picn(X) with Pic0(X) via the invertible sheaf X(−nP ).)Then we see that i∗(Ber(gr)) p∗(Ber(Pic0(X))). This fact allows us to proveTheorem 2.14.1.

Note first that we have a surjective map

Heis− → H 1(X,X).(4.5)

Indeed, let X = U 0 ∪U P be an open cover where U 0 = X −P red and U P is a suitabledisk around P red. Then if [a] ∈ H 1(X,X) is represented by a ∈ X(U 0 ∩ U P ) wecan write, using the local coordinates on U P , a = aP +

i>0 ai z−i + αi z−i θ , with

aP ∈ X(U P ). Then a − aP =

ai z−i + αi z−i θ ∈ Heis− and [a] = [a − aP ]. Nowthe tangent space to any point ∈ Pic0(X) can be identified with H 1(X,X) and sowe have a surjective map from Heis− to the tangent space of Pic0(X). Note secondlythat a change of trivialization of , given by t → t , corresponds to multiplicationof the point W ∈ gr by an element a0 + α0θ +

i>0 ai zi + αi zi θ of the group

corresponding to Heis+. From these two facts we conclude that there is a surjective

map from Heis to the tangent space to the image of the Krichever map i : M →grat any point W = W(X,P,(z,θ),, t). Now the first Chern class of Ber(gr) iscalculated from the cocycle (3.5) on gl∞|∞() and it follows from Lemma 3.3.1 thatthe restriction of this cocycle to Heis is identically zero. This implies that

i∗

c1[Ber(gr)]

= p∗

c1

Ber

Pic0(X)

= 0.

Butthemap p : M → Pic0(X) is surjective, so we finally findthat c1(Ber(Pic0(X))) =0 and Ber(Pic0(X)) is topologically trivial, proving Theorem 2.14.1.

4.3. Algebro-geometric τ - and Baker functions. We consider geometric data map-ping to W in the big cell of gr, so that deg = g − 1. As discussed in Section 3,we can associate to W both a τ -function and a Baker function. A system of superKP flows on gr applied to W produces an orbit corresponding to a family of defor-mations of the original geometric data. The simplest system of super KP flows, the“Jacobian” system of Mulase and Rabin [Mu], [R1], deforms the geometric data bymoving in Picg−1(X). Solutions to this system for X a superelliptic curve wereobtained in terms of super θ-functions in [R2]. On the basis of the ordinary KP theory(cf. [SW, Section 9]), we might expect that in general the τ - and Baker functions forthis family can be given explicitly as functions of the flow parameters by means of the super θ-functions (when these exist) on the Jacobian of X. We now discuss theextent to which this is possible.

Recall from (4.5) that we have a surjection from Heis−,ev to the cohomologygroup H 1(X,X,ev). By exponentiation we obtain a map from Heis−,ev to Pic0(X),

and these maps fit together in a diagram:




0

0

H 1(X,Z)

0 / / K0

/ / Heis−,ev / / H 1(X,X,ev)

/ / 0

0 / / K

/ / Heis−,ev / / Pic0(X)

/ / 0.

K/K0

0

0

(4.6)

Here K0 is the ev-submodule of elements f of Heis−,ev that split as f = f 0 +f P ,with f 0 ∈ X(U 0) and f P ∈ X(U P ),and K is theAbelian subgroup (not submodule!)of elements k of Heis−,ev that after exponentiation factorize; that is, ek = φkekp ,with φk ∈ X(U 0)× and kP ∈ X(U P ). From the Snake lemma it then follows thatH 1(X,Z) K/K0. So a function F on Heis−,ev descends to a function F onH 1(X,X,ev) if it is invariant under K0. The automorphic behaviour of such a functionF with respect to the lattice H 1(X,Z) translates into behaviour of F under shifts byelements of K. In particular we consider the function τ W associated to a point W

in the big cell of gr; see (3.14) and (3.15). This is a function on Heis−,ev, and,because of Lemma 3.3.1, we see by an easy adaptation of the proof of Lemma 9.5 in[SW] that

τ W (f + k) = τ W (f)τ W (k), f ∈ Heis−,ev, k ∈ K.

In particular we obtain by restriction a homomorphism

τ W : K0 → ×ev.

Let η : K0 → ev be a homomorphism such that τ W (k0) = eη(k0), for all k0 ∈ K0.Then we can define a new function

τ 1(f ) = τ W (f)e−η(f).

Then τ 1(k0) = 1, but still we have

τ 1(f + k) = τ 1(f )τ 1(k),(4.7)



SUPERCURVES 47

so that τ 1 descends to a function τ 1 on H 1(X,X,ev). From (4.7) we see that τ 1corresponds to a (meromorphic) section of a line bundle on Pic0(X) with trivialChern class.

A suitable ratio of translated θ-functions gives a section of this same bundle, so that

τ 1 is expressed as this ratio times a meromorphic function, the latter being rationallyexpressible in terms of super θ-functions. Then the modified τ -function τ 1 is rationallyexpressed in terms of super θ-functions.

The even Baker function wW

0(z,θ) associated to the point W is just the even section

of holomorphic except for a pole 1/z at P . Such a section can be specified by itsrestrictions to the charts U 0 and U P . The Jacobian SKP flows act by multiplying thetransition function of across the boundary of U P by a factor γ(t) as in (3.13). Thecorresponding action on the associated point W of gr is generated by the matricesλ(n) + µ(n) and f(n) of Section 3; the remaining matrices generate deformations of the curve X and enter the Kac–van de Leur SKP flows. Then w

W(t)

0/wW

0is a section of

the bundle with transition function (3.13). Equivalently, it is a meromorphic functionon U 0 that extends into U P except for an essential singularity of the form (3.13),having zeros at the divisor of (t ) and poles at the divisor of . By analogy with the“Russian formula” of ordinary KP theory, such a function would be expressed in theform

exp

∞k=1

(z,θ)

(0,0)

t k ψk + t k−(1/2)Ek

+ c(t)

(4.8)

times a ratio of θ-functions providing the zeros and poles. Here ψk and Ek aredifferentials on X, with vanishing a periods and holomorphic except for the behaviournear P ,

ψk ∼ D

z−k

= −kρz−(k+1), Ek ∼ D

θ z−k

= z−k.(4.9)

The constant c(t) is linear in the flow parameters. In addition to the symmetry of theperiod matrix, we have to require the existence of these differentials. This requiresthat they exist in the split case, and then that these split differentials extend throughthe sequence (2.21). In the split case, the odd differentials ψk are just θ times theordinary differentials on the reduced curve, which appear in the Russian formula (andthey do extend). However, the even differentials Ek are sections of , which is of degree zero and nontrivial, with h1 = g − 1. Consequently, when g > 1 there areWeierstrass gaps in the list of pole orders of these differentials. This means that theodd flow parameters corresponding to the missing differentials must be set to zeroin order for the Baker function to assume the Russian form. Even then, however,the function given by the Russian formula generically behaves as 1 + αθ +(z) forz → 0, rather than the correct 1 + O(z) for w

W(t)

0containing no θ/z pole. In [R3]

this is dealt with by including a term ξ E0 in the exponential, taking ∂ξ to construct




a section with a pure θ/z pole, and subtracting off the appropriate multiple of this.In general, however, no such E0 will exist. These difficulties are understandable inview of the relations (3.17), which require that the τ -function be known for the fullset of Kac–van de Leur flows in order to compute the Baker functions for even the

Jacobian flows. Since the dependence of the τ -function on the non-Jacobian flows islikely to be far more complicated than our super θ-functions, it is unlikely that theBaker functions can be expressed in terms of them.

Appendices

A. Duality and Serre duality. Let be our usual ground ring C[β1, . . . , βn]with the βi odd indeterminates. In this appendix we discuss duality for -modules(cf. Chapter 21 in Eisenbud [E] for the case of commutative rings). Then we use thisto extend Serre duality for supermanifolds over the ground field C (refer to [HW],[OP]) to supermanifolds over the ground ring .

Finally we discuss the more explicit form of Serre duality that one has in the caseof supercurves. One way of proving Serre duality for supercurves over general

would be to use the properties of the supertrace in infinite-rank -modules to definea residue, generalizing the method of Tate [Ta]. However, we have available in ourcase contour integration which also provides us with a residue map.

A.1. Duality of -modules. Let M be an object of od, the category of finitelygenerated -modules. We give the Z2-graded vector space E(M) = HomC(M,C)

the structure of a -module via

λ · φ(m) = (−1)|λ||φ|φ(λm),

for all m ∈ M and all homogeneous λ ∈ and φ ∈ E(M). Then one checks thatE = E(−) is a dualizing functor on od: it is contravariant, is exact, is -linear,and satisfies E2 1od . One also checks that E() , up to a possible paritychange. In the sequel we will ignore these parity changes.

An explicit (basis-dependent) isomorphism → HomC(,C) can be describedas follows. The monomials βI = β

i11 · · · β

inn , ij = 0, 1, form a basis of as a C-

vector space, and we let φβ I be the dual basis of E() = HomC(,C), so thatφβ I (βJ ) = δI J . Then the -homomorphism that maps 1 ∈ to the linear functionalφβ1···βn is an isomorphism, odd in the case when n is odd.

From the fact that E is a dualizing functor we see that the map Hom(M,N) →Hom(E(N),E(M)) is an isomorphism for all objects M, N and hence

E(M) = Hom

,E(M)

Hom

M,E()

= Hom(M,).

In other words, up to a possible parity switch, we can identify (functorially) the-modules of C-linear and of -linear homomorphisms on any finitely generated-module M ,

HomC(M,C) Hom(M,).



SUPERCURVES 49

We identify the two and use for both the symbol M ∗.Note that the above implies that the functor that maps M to M ∗ = Hom(M,) on

od (naturally isomorphic to the functor E) is exact: exact sequences get mappedto exact sequences. Equivalently, is injective as a module over itself. Also note that

the double dual of M is isomorphic to M itself. A.2. Serre duality of supermanifolds. Serre duality for supermanifolds (Y,Y ) of

dimension (p | q) over the complex numbers is discussed in Haske and Wells [HW]and in Ogievetsky and Penkov [OP].

A dualizing sheaf on (Y,Y ) is an invertible sheaf ωY together with a fixed homo-morphism

t : H p

X, ωY

→C

such that the induced pairing for all

H i

Y,∗ ⊗ ωY

⊗ H p−i (Y,)

t −→C

gives an isomorphism, called Serre duality,

H i

Y,∗ ⊗ ωY

∼−→ H p−i (Y,)∗.(A.1)

Note that this is an isomorphism of Z2-graded vector spaces overC. Dualizingsheavesare unique, up to isomorphism.

If Y is the tangent sheaf of (Y,Y ) with transition functions J αβ : U α ∩ U β →Gl(p | q,C), then the Berezinian sheaf erY is the invertible sheaf with transitionfunctions ber(J αβ) : U α ∩ U β → C×. In [HW], [OP], it is proved that erY is adualizing sheaf.

We are interested in the relative situation: supermanifolds (Z,Z ) → (•, ) of dimension (p | q) over . The structure sheaf Z contains n independent globalodd constants. We can reduce this to the absolute case by thinking of (Z,Z ) as asupermanifold (Z,Y ) of dimension (p | q + n) over C: the constants βi generating

are now interpreted as coordinates.The tangent sheaf Y of derivations of Y has then global sections ∂/∂βI , in

contrast to the relative tangent sheaf Z/. One sees easily, however, that the in-vertible sheaf erY , constructed from the tangent sheaf Y , is the same as erZ/,constructed from the relative tangent sheaf Z/.

Now let be a coherent, locally free sheaf of Z-modules. Then the associatedcohomology groups are finitely generated -modules, so we can use the theory of Appendix A.1. In particular we see that there is a homomorphism

t : H p

Z,erZ/

→ (A.2)

inducing an isomorphism of -modules, also called Serre duality,

H i

Z,∗

⊗erZ/ ∼

−→ H p−i

(Z,)∗

,(A.3)




where now H p(Z,)∗ means the -linear dual.In case of N = 1, 2 supercurves (X,X) → (•, ) we can be more explicit about

the homomorphism t in (A.2). Write erX for the relative Berezinian erX/. Thecohomology of erX is calculated by the sequence

0 → H 0

X,erX

→ RaterX

→ Prin

erX

→ H 1

X,erX

→ 0,

where Rat(erX) are the rational sections and Prin(erX) the principal parts. OnPrin(erX) we have for every x ∈ X a residue map Resx ω → (1/2π i)

Cx

ω, whereCx is a contour around x, using the integration on N = 1, 2 curves introduced inSubsection 2.4. Then t =

x∈X Resx and the classical residue theorem holds: if

ω ∈ Rat(erX) then t(ω) = 0. Now that we have the residue we can proceed toproving Serre duality as in the classical case; cf. [Se].

B. Real structures and conjugation. Let = C[β1, . . . , βn] be the Grassmannalgebra generated by n odd indeterminates. Choose a sign = ±1 and define a realstructure on for this choice as a real-linear, even map ω : → such that

1. ω(ca) = cω(a),c ∈C

, a ∈ ,2. ω2 = 1,3. ω(βi ) = βi ,4. ω(ab) = |a||b|ω(b)ω(a),

where a, b are homogeneous elements of parity |a|, |b|. We often write a for ω(a).A real structure induces a decomposition of into eigenspaces, = ⊕ ,

where is the +1 eigenspace and the −1 eigenspace for ω. Multiplication byi is an isomorphism, so we get = ⊕ i.

C. Linear equations in the super category

C.1. Introduction. Cramer’s rule tells us that a system of linear equations over acommutative ring R,

xA = y,(C.1)

with x = (x1, . . . , xn) and y = (y1, . . . , yn) n-component row vectors and A an n × n

matrix with coefficients in R, can be solved by quotients of determinants:

xi =det

Ai (y)

det(A), i = 1,. . . ,n ,(C.2)

where Ai (y) is the matrix obtained by replacing in A the ith row by the row vector y.We want to study (C.1) in the super category. So we fix a decomposition n = k + l.

Call an index i ∈ {1, . . . , n} even if i ≤ k and odd otherwise. Consider an even matrixA = (aij )n

i,j =1 over : aij is an element of the even part ev if i and j have the sameparity and of the odd part odd otherwise. Note that it is not necessary to specify the

parity of y in (C.1); it can be even, odd, or inhomogeneous.



SUPERCURVES 51

In the theory of linear algebra over a Grassmann algebra in many (but not all)respects, a role analogous to that of the determinant in the commutative case is playedby the Berezinian and its inverse: for an even matrix A =

X αβ Y

we define

ber(A) = det X − αY −1βdet Y −1,

ber∗(A) = det

X−1det

Y − βX−1α

=1

ber(A).

(C.3)

We discuss how Cramer’s rule can be generalized in this setting. The result is asfollows: The solution of (C.1) for A an even supermatrix is given by

xi =

ber

Ai (y)

/ber(A), if i ≤ k,

ber∗

Ai (y)

/ber∗(A), if i > k.(C.4)

Note that in general the matrix Ai (y) is not even. However, in ber(A) for i ≤ k

(respectively, ber∗(A) for i > k) the entries aij , j = 1, . . . , n of row i occur onlylinearly. So, if ber(A) (respectively, ber∗(A)) exists, we mean by ber(Ai (y)) fori ≤ k (respectively, ber∗(Ai (y)) for i > k) the element of obtained by replacing in

ber(A) (respectively, ber∗(A)) all aij by yj for j = 1, . . . , n.The expression for the entries in even positions, that is, xi , i ≤ k, can be found in

[UYI], but we have not seen the solution for the odd positions in the literature. Themain ingredient is the Gelfand-Retakh theory of quasideterminants [GR1], [GR2].

C.2. Quasideterminants. Let A = (aij )ni,j =1 be an n × n matrix with entries be-

ing independent (noncommuting) variables aij . Let Aij be the submatrix obtainedby deleting in A row i and column j . Following Gelfand-Retakh we introduce n2

quasideterminants |A|ij , rational expressions in the variables apq . If n = 1 we put|A|11 = a11 and for n > 1 we define recursively

|A|ij = aij −

p=j

q=i

aip

Aij

−1qp

aqj .(C.5)

If we assume that the variables aij commute amongst themselves, then we have

|A|ij = (−1)i+j det(A)/det

Aij

.(C.6)

Returning to the general case, one proves that the inverse of A, as a matrix over thering of rational functions in the aij , is the matrix B = (bij ), where bij = |A|−1

j i . Thismeans that we can also define the quasideterminant as

|A|ij = aij −p=j

q=i

aipc(ij)pq aqj ,(C.7)

where C(ij) = (c(ij)pq ) is the inverse to the matrix Aij . This second definition is useful

if one wants to think of the entries of A as elements of a ring R. In that case it is




perfectly possible that the inverse C(ij) of Aij exists, but that some entry c(ij)pq is not

invertible in R. In this situation (C.7) allows us to define the quasideterminant |A|ij ,whereas (C.5) might make no sense.

Quasideterminants have the following properties.

P1. If the matrix B is obtained from A by multiplying row i from the left by λ (anew independent variable), then for all j = 1, 2, . . . , n we have

|B|ij = λ|A|ij , |B|kj = |A|kj , k = i.

Similarly if the matrix C is obtained from A by multiplying column j from the rightby µ we have for all i = 1, . . . , n,

|C|ij = |A|ij µ, |C|ik = |A|ik , k = j.

P2. If B is obtained from A by adding row k to some other row, then for allj = 1, . . . , n,

|B|ij = |A|ij , k = i.

If C is obtained from A by adding column k to some other column, then for alli = 1, . . . , n,

|C|ij = |A|ij , k = j.

C.3. Restriction to superalgebra. We keep the decomposition n = k + l and weconsider an even matrix A = (aij )n

i,j =1 over supercommuting variables: aij is an evenvariable if i and j have the same parity and an odd variable otherwise.

Lemma C.3.1. Let A be an even matrix as above.

• If i, j are both even, then

|A|ij = (−1)i+j ber(A)/ber

Aij

.

• If i, j are both odd, then

|A|ij = (−1)i+j ber∗(A)/ber∗

Aij

.

Proof. We can write

A =

X α

β Y

=

1 αY −1

0 1

X − αY −1β 0

0 Y

1 0

Y −1β 1

.

This shows that A is obtained from

X−αY −1β 00 Y

by operations that do not change

the quasideterminant |A|ij in the case when i, j are both even. Since the last matrixhas entries that commute with each other we can use (C.6) to find

|A|ij = (−1)i+jdetX − αY −1β

det

[X − αY −1β]ij .(C.8)



SUPERCURVES 53

Now for i, j both even [X − αY −1β]ij = Xij − αi∅Y −1β∅j , where αi∅ is the matrixobtained by deleting just the row i and β∅j is obtained by deleting column j . Sodet([X − αY −1β]ij ) det(Y −1) = ber(Aij ) and the first part of the lemma follows bymultiplying the numerator and denominator of (C.8) by det(Y −1). For the second part

use the decompositionA =

1 0

βX −1 1

X 00 Y − βX−1α

1 X−1α

0 1

.

C.4. Linear equations. Now we think of the matrix A = (aij )ni,j =1 as an even

matrix over . Then the quasideterminants |A|ij , thought of as elements of , canonly exist if i, j have the same parity (since otherwise Aij is never invertible). Evenif i, j have the same parity |A|ij may or may not exist (as an element of ), but if it does then the quasideterminant |Ai (y)|ij exists also. (Recall that Ai (y) is obtainedby replacing the ith row of A by the row vector y = (y1, . . . , yn).) Indeed, accordingto (C.5), if y = (y1, . . . , yn), then, using that Ai (y)ij = Aij ,

|Ai (y)|ij = yi −

p=i, q=jypA

ij −1qp aqj .

Hence, using Lemma C.3.1, we find, whenever |A|ij exists,

|Ai (y)|ij =

(−1)i+j ber

Ai (y)

/ ber

Aij

, if i ≤ k,

(−1)i+j ber∗

Ai (y)

/ ber∗

Aij

, otherwise.(C.9)

Now consider the linear system

xA = y,(C.10)

with x, y row vectors and A an even matrix over . Using the properties of thequasideterminant, (C.10) implies

xi |A|ij = |Ai (y)|ij .

Combining (C.9) and Lemma C.3.1 proves (C.4).Of course, if one deals with infinite systems of equations (C.1) over a supercom-

mutative ring, one has the same expressions (C.4) for the solutions, provided that onecan define a Berezinian (with the usual properties) of the infinite matrices involved.

D. Calculation of a super τ -function. The Baker functions for arbitrary linebundles over a superelliptic curve were computed in [R3]. Note that a superellipticcurve, meaning an N = 1 SRS of genus 1 with trivial spin structure is not ageneric SKP curve and its Jacobian is not a supermanifold. Still, the correspondingsuper τ -function can be computed using the Baker-τ relations (27,28) of [DS]:

ber

B(W,F,S ∗) · S ∗

=

τ

W, F T 1− Sz−1

−1/τ

W, F T −1

,(D.1)




in the notation of that paper. We choose the 2 ×2 matrix S to be diag(ζ,ζ), with ζ aneven variable; then the entries of the 2 × 2 matrix Baker function Bij (W,F,S ∗) aresimply the coefficients of the even Baker function w0 = B00(ζ ) + B01(ζ)θ and theodd Baker function w1 = B10(ζ ) + B11(ζ)θ for the subspace F W of gr, where F

is multiplication by some invertible formal Laurent series f(z) + φ(z)θ, viewed as a2×2 matrix

f 0φ f

. This action of F corresponds via the Krichever map to deforming

by tensoring it with a bundle whose transition function across the circle |z| = 1 isf(z) + φ(z)θ.

The results of [R3] give the entries of the Baker matrix as

B00 = 1 +αδ

2π i

(a)(ζ − a)

(a)(ζ − a)+

(a)2

(a)2

,(D.2)

B01 = α(a)

(a)+ terms proportional to δ,(D.3)

B10 =δ

2π i (ζ − a)

(ζ − a)+

(a)

(a) ,(D.4)

B11 = 1 +αδ

2π i

(a)

(a)−

(a)2

(a)2−

(ζ − a)

(ζ − a)+

(ζ − a)2

(ζ − a)2

.(D.5)

Here τ and δ are the even and odd moduli of the superelliptic curve X; all θ-functions appearing are 11(•; τ ). The parameters (a,α) label the point in the superJacobian corresponding to the deformed bundle ; if F is written in the form

F = exp∞

i=1

t i z−i + t i−(1/2)θ z−i

,(D.6)

then they are linear combinations of the t i plus constants labeling itself.Computing the superdeterminant, we find

τ

W, F T 1 − ζ z−1

−1τ

W, F T =

1 −αδ

2πi[log(a − ζ )]

1 −αδ

2π i[log(a)]

.(D.7)

The function

1 − ζ z−1−1

= exp∞

k=1

1

k

ζ k

zk(D.8)

produces the usual shifts by ζ k /k in the even parameters t k . As a transition functionfor a line bundle, z/(z − ζ ), it corresponds to the principal divisor (0, 0) − (ζ, 0), sothe corresponding deformation adds −ζ to the even coordinate a of the Jacobian, as

we see on the right side of (D.7). We conclude that



SUPERCURVES 55

τ

W, F T

= 1 −αδ

2π i

log (a)

.(D.9)

Note, however, that if the 2 × 2 matrix F represents multiplication by f(z) + φ(z)θ,then F T represents the action of f(z) +φ(z)∂θ , which in terms of the Krichever data

is no longer just a deformation of , but of X as well. That is, the Baker functionfor the orbit of F enables us to calculate the τ -function for the “dual” orbit of F T , atotally different type of deformation.

We see also that, consistent with the discussion in Subsection 4.3, the super τ -function is a genuine function, not a section of a bundle, as the multivaluedness of the θ-function is eliminated by the logarithmic derivatives. This is again due to thefact that the restricted group of F T ’s considered here acts without central extensionin the Berezinian bundle.

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Bergvelt: Department of Mathematics, University of Illinois, Urbana, Illinois 61801,USA; [email protected]

Rabin: Department of Mathematics, University of California at San Diego, La Jolla,

California 92093, USA; [email protected]

m.j. bergvelt and j.m. rabin- supercurves, their jacobians, and super kp equations

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