arxiv.org · skt geometry gil r. cavalcanti department of mathematics utrecht university abstract...
TRANSCRIPT
SKT geometry
Gil R. Cavalcanti∗
Department of MathematicsUtrecht University
Abstract
SKT structures are closely related to Kahler structures, the difference being that inthe Kahler case one has a complex structure which is parallel with respect to the Levi–Civita connection, while in the SKT the complex structure is parallel with respect to ametric connection with skew-symmetric and closed torsion, a concept which is a little morerestrictive than asking that such connection has holonomy in U(n). The inclusion of thetorsion, however leaves several of the usual arguments used in Kahler geometry without adirect counterpart. We use tools from generalized complex geometry to develop the theoryof SKT manifolds. We develop Hodge theory on SKT manifolds and, more generally, onparallel Hermitian manifolds and prove that their twisted cohomology inherits a Z × Z2-grading determined by the structure. We study Lie algebroids and differential operatorsassociated to SKT structures and study the deformation theory of these structures. Asapplications we reobtain a result of Lubke and Teleman regarding the existence of SKTstructures on the moduli space of instantons of a bundle over a complex surface and showthat even though Kahler structures are not stable under deformations of the symplecticstructure, small deformations are still SKT.
MSC classification 2010: 53C25; 53D18; 53C15.Subject classification: Differential geometry.Keywords: strong KT structure, generalized complex geometry, generalized Kahler geom-etry, Hodge theory, deformations.
Contents
1 Linear algebra 5
2 Intrinsic torsion of generalized Hermitian structures 112.1 The Nijenhuis tensor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122.2 The intrinsic torsion and the road to integrability . . . . . . . . . . . . . . . 14
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3 SKT structures 16
4 Parallel Hermitian and bi-Hermitian structures 21
4.1 Parallelism and the intrinsic torsion . . . . . . . . . . . . . . . . . . . . . . 21
5 Hodge theory 26
5.1 Differential operators, their adjoints and Laplacians . . . . . . . . . . . . . 26
5.2 Hodge theory on SKT manifolds . . . . . . . . . . . . . . . . . . . . . . . . 28
5.3 A spectral sequence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30
5.4 Relation to Dolbeault cohomology . . . . . . . . . . . . . . . . . . . . . . . 33
5.5 Alternative definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36
6 Instantons over complex surfaces 39
7 Lie algebroid differentials and other operators 42
7.1 Lie algebroids . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42
7.2 Further operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44
7.3 Representations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49
8 Deformations 51
8.1 First deformation problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52
8.2 Second deformation problem and stability . . . . . . . . . . . . . . . . . . . 53
8.3 Deformations of generalized Kahler structures . . . . . . . . . . . . . . . . . 59
Introduction
Looking beyond the Levi–Civita connection in Riemannian geometry, one finds a numberof other metric connections with interesting properties. Normally these families of con-nections are defined by some characteristic of the torsion tensor and recurrent themes ofresearch are connections with parallel torsion or connections with skew symmetric torsion.The “strong” torsion condition refers to the last group of connections: a strong torsionconnection on a Riemannian manifold is a metric connection whose torsion is skew sym-metric and closed. In this setting, a Kahler structure with strong torsion, or SKT structureis a Hermitian structure (g, I) together with a strong torsion connection for which I isparallel. A weaker notion is that of a (strong) parallel Hermitian structure which for usmeans a connection with closed, skew symmetric torsion for which the holonomy is U(n).The difference between a parallel Hermitian structure and an SKT structure being thatfor parallel Hermitian structures, integrability of the complex structure is not required.
A reason to study of such objects comes from string theory, where closed 3-forms arisenaturally as fields in their sigma models [30, 31]. Once a 3-form is added to the sigmamodel, if one still requires a nontrivial amount of supersymmetry, the type of geometryof the target space has to move away from the usual Kahler geometry. It was preciselyfollowing this path that Gates, Hull and Rocek [14] discovered the bi-Hermitian geometrythat nowadays also goes by the name of generalized Kahler geometry [20] as the solutionsto the (2, 2)-supersymmetric sigma model. Requiring less supersymmetry without giving
2
up on the idea altogether leads one to consider models where there is more left than rightsupersymmetry or models where the right side is simply absent. These conditions leadto (2, 1) or (2, 0) supersymmetric sigma models and supersymmetry holds if and only ifthe target space has an SKT structure [23]. This point of view also leads one to considerparallel Hermitian and bi-Hermitian structures as these are geometric structures imposedby a sigma model with an extended supersymmetry algebra [9, 10].
There is yet another simple mathematical description of SKT structures: they corre-spond to Hermitian structures (g, I) for which the Hermitian form ω = g(I·, ·) satisfies
ddcω = 0,
where dc = i(∂ − ∂). The 3-form H = dcω is the torsion of the connection mentionedearlier.
Since SKT manifolds are, in particular, complex, their space of forms inherits a naturalbi-grading, but a simple check in concrete examples shows that there is no correspondingdecomposition of their cohomology. Further, by studying these structures on six dimen-sional nilmanifolds Fino, Parton and Salamon [12] produced examples showing that theirFrolicher spectral sequence does not necessarily degenerate at the second page, they donot satisfy the ddc-lemma, manifolds carrying these structures may not be formal and thatthese structures are not stable under deformations of the complex structure. In short,several Kahler properties seem to have been lost once the torsion was included and nowa-days much of the research in this area revolves around the search of examples of manifoldsadmitting SKT structures.
In fact, until now there were very few positive, conceptual results about SKT manifoldsand a feature of these results was that they took the torsion 3-form into account in asubstantial way. Remarkably, using the connection with closed 3-form torsion, Bismutproved a local index theorem for SKT manifolds in [2], extending previous results knownfor Kahler manifolds. Studying a slightly different problem, Gauduchon showed in [15]that any compact complex surface admits one SKT metric on each conformal class ofHermitian metrics, a result which proved to be fundamental in the study of non Kahlercomplex surfaces as carried out by Lubke and Teleman [28]. In particular, Lubke andTeleman proved that given a vector bundle over a complex Hermitian surface (with itsonly SKT structure), the smooth locus of the moduli space of instantons inherits an SKTstructure [28] and the presence of torsion means that the metric on the moduli space is notthe usual quotient metric, but something modeled on spaces transverse, but not orthogonalto the orbits. This phenomenon was re-encountered by Grantcharov, Papadopolous andPoon [18] when extending symplectic quotients to the SKT world. Indeed, there theyobserved that the torsion form forced them to choose two moment maps: the first cuts asubmanifold, as usual, and the second guides on the choice of metric model for the quotient,which, once again, is not the orthogonal complement of the group orbits.
The last two examples were unified by the author in [7] where they were also put in theframework of reduction of Courant algebroids and generalized actions introduced in [3].The key observation was that SKT structures have yet another description, this time, asa ‘generalized structure’, i.e., a geometric structure on TM = TM ⊕ T ∗M . In fact, in thispaper we show that, in a very precise way, SKT structures lie half way between generalizedHermitian and generalized Kahler structures. Given the success generalized geometry has
3
had in the treatment of bihermitian structures, it is only natural to ask if it can do thesame for SKT structures. In this paper we show that to be the case. In fact we showthat it provides the correct point of view to study SKT structures and that several of thenegative results mentioned earlier have a positive counterpart involving the torsion.
Indeed, the first observation is that, as structures defined on TM with the H-Courantbracket, the natural differential operator to consider is dH = d+H∧, where H = dcω, andω is the Hermitian form. Hence questions about d, and its decomposition as ∂ + ∂ missan important ingredient and were doomed from the start. The cohomology of dH is onlyZ2-graded, yet we show that a parallel Hermitian structure induces a Z×Z2 grading on thespace of forms which itself induces a Z×Z2 grading on the dH -cohomology. This is achievedby introducing the intrinsic torsion of a generalized almost Hermitian structure and usingit to show that the dH -Laplacian preserves a Z×Z2-graded decomposition of the space offorms. For SKT manifolds one can go further and prove an identity of Laplacians, just asMichelsohn did for Kahler manifolds [29] and, more recently, Gualtieri, in the generalizedKahler context [19]. This way we relate a cohomology naturally defined in terms of theSKT data with the dH -cohomology. These identities imply further that a variation of theFrolicher spectral sequence degenerates at the second page. As an application, we returnto the moduli space of instantons on a bundle over a compact complex surface and showthat the existence of an SKT structure in this space can be seen as a consequence of theHodge theory developed for SKT manifolds.
Following the development of generalized complex geometry we study natural Lie al-gebroids associated to an SKT structure. In this case, since an SKT structure is only‘half’ generalized Kahler, several of the bundles which are Lie algebroids in the generalizedKahler case cease to be in the SKT set up. Yet, one can still find several interesting oper-ators similar to the Lie algebroid differentials but associated to subbundles of TM whichare not involutive. These operators re-appear in our study of the problem of deformationsof SKT structures. Precisely, they appear in the obstructions to deformations. Given theclose proximity between SKT and generalized Kahler structures, we study the problemof deformations of generalized Kahler structures. Specifically, we show that the problemof deformations of generalized Kahler structures is obstructed and find the cohomologyspace where the obstructions lie. As an application of the deformation theory, we finishthe paper showing that if (M,ω, I) is a compact Kahler manifold then deformations of thesymplectic structure ω can always be completed with deformations of I into an generalizedalmost complex structure such that the pair of deformed structures is an SKT structure.In contrast, there may be no deformation of I which together with the deformed symplecticstructure forms a (generalized) Kahler structure.
This paper is organized as follows. In Section 1 we develop the linear algebra pertinentto generalized complex, generalized Hermitian and SKT structures. In particular we showthat an SKT structure gives rise to a Z × Z2-grading on the space of forms. In Section 2we introduce the intrinsic torsion of a generalized Hermitian structure and in Sections 3and 4 relate SKT structures and parallel Hermitian structures to the vanishing of certaincomponents of the intrinsic torsion. In particular, we show that for an integrable SKTstructure the operator dH decomposes into a sum of three operators defined in terms ofthe SKT data, while for parallel Hermitian structures it decomposes as a sum of fiveoperators. In the presence of a generalized complex extension of the SKT structure, dH
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decomposes further, into six operators. In Section 5 we study Hodge theory for SKT andparallel Hermitian structures and prove that in both cases the dH -cohomology decomposesaccording to the decomposition of forms induced by the structure. As an application, inSection 6 we recover the result that the moduli space of instantons over a complex surfacehas an SKT structure. In Section 7 we study Lie algebroids, Lie algebroid differentials andsimilar operators defined on an SKT manifold and in Section 8 we use these operators tostudy the problem of deformations of SKT and generalized Kahler structures.
Acknowledgements: The author would like to thank Ulf Lindstrom, Martin Rocek,Stefan Vandoren and Maxim Zabzine for helpful input regarding the relations to super-symmetric sigma models. This research was supported by a Marie Curie Intra EuropeanFellowship.
1 Linear algebra
Given a vector space V m we let V = V ⊕ V ∗ be its “double”. V is endowed with a naturalsymmetric pairing:
〈X + ξ, Y + η〉 =1
2(η(X) + ξ(Y )), X, Y ∈ V ξ, η ∈ V ∗.
Elements of V act on ∧•V ∗ via
(X + ξ) · ϕ = iXϕ+ ξ ∧ ϕ.
One can easily check that for v ∈ V
v · (v · ϕ) = 〈v, v〉ϕ,
hence ∧•V ∗ is naturally a module for the Clifford algebra of V. In fact, it is the space ofspinors for Spin(V) and hence comes equipped with a spin invariant pairing, the Chevalleypairing:
(ϕ,ψ)Ch = −(ϕ ∧ ψt)top,
where ·t indicates transposition, an R-linear operator defined on decomposable forms by
(α1 ∧ · · · ∧ αk)t = αk ∧ · · · ∧ α1,
and top means taking the degree m component.The spin group, Spin(V), acts on both V and on spinors in a compatible manner,
namely, its action on V is by conjugation using Clifford multiplication
g∗v = gvg−1 for all g ∈ Spin(V), v ∈ V
and on ∧•V ∗ by the Clifford action described above, so we have
g(v · ϕ) = (gvg−1)(gϕ) = (g∗v)(g · ϕ) for all g ∈ Spin(V), v ∈ V and ϕ ∈ ∧•V ∗.
We will be particularly interested in the actions of 2-forms and 2-vectors, which wedescribe below to fix signs and notation.
5
Example 1.1. Given B ∈ ∧2V ∗ ⊂ spin(V), it acts on V. Namely, we can regard B as amap from V into V ∗
B(X + ξ) = −iXB = −B(X).
Exponentiating this map we have
eB∗ : V −→ V eB∗ (X + ξ) = X + ξ −B(X).
And this action is compatible with the action of eB ∈ Spin(V) on forms:
eB : ϕ −→ eB ∧ ϕ.
So we have eB∗ : ∧kV −→ ∧kV and for α ∈ ∧•V and ϕ ∈ ∧•V ∗
(eB∗ α) · eBϕ = eB(α · ϕ).
Example 1.2. Similarly, given β ∈ ∧2V we get maps
β(X + ξ) = iξβ = −β(ξ),
so for example if β = ∂x1 ∧ ∂x2 and ξ = dx1 then β(ξ) = ∂x2 . Exponentiating, we get
eβ∗ : V −→ V eβ∗ (X + ξ) = X + ξ − β(ξ).
And we have a corresponding action on spinors
eβ : ∧•V ∗ −→ ∧•V ∗ eβ : ϕ −→ eβ · ϕ,
where the dot indicates interior product. The convention for interior product by a multi-vector here is such that ∂x1 ∧ ∂x2 · dx1 ∧ dx2 = −1.
These two actions are compatible:
(eβ∗α) · eβϕ = eβ(α · ϕ) for all α ∈ ∧•V, ϕ ∈ ∧•V ∗. (1.1)
We will be interested in introducing geometric structures on V. The first we consideris a generalized metric, as introduced by Gualtieri [20].
Definition 1.3. A generalized metric on V is an automorphism G : V −→ V which isorthogonal and self-adjoint with respect to the natural pairing and for which the bilineartensor
〈Gv, w〉, v, w ∈ V
is positive definite.
Since G is orthogonal and self-adjoint, we have G−1 = Gt = G, hence G2 = Id. ThereforeG splits V into its ±1-eigenspaces: V = V+ ⊕ V− and the projection πV : V −→ Vgives isomorphisms π : V± −→ V . Further, given a generalized metric G we can writeV = GV ∗⊕V ∗ and GV ∗ is isomorphic to V via the projection πV : V −→ V . Since both Vand GV are isotropic subspaces of V which project isomorphically onto V , we can describeGV as the graph of a linear map b : V −→ V ∗, that is b ∈ ⊗2V ∗. Isotropy means that
6
b ∈ ∧2V ∗ and hence gives rise to an orthogonal transformation of the natural pairing, eb∗.This map has the property that eb∗ : GV ∗ −→ V , hence, after an orthogonal transformationof V, we can assume that GV ∗ = V . For this splitting,
G =
(0 gg−1 0
)where g is an ordinary metric on V . The splitting of V determined by a generalized metricis the metric splitting.
If V is endowed with an orientation, we can define a generalized Hodge star operator [19]as follows. Since πV : V+ −→ V is an isomorphism, V+ inherits an orientation. Then we lete1, e2, · · · , em be a positive orthonormal basis of V+ and let ? = −em · · · · e2 ·e1 ∈ Clif(V).Then
?ϕ := ? · ϕ,
where · denotes Clifford action.With this definition, we have
(ϕ, ?ϕ)Ch > 0 if ϕ 6= 0
If the splitting of V is the metric splitting, we have
(ϕ, ?ψ)Ch = ϕ ∧ ∗ψ,
where ∗ is the usual Hodge star, hence, in this splitting, ? is the usual Hodge star exceptfor a change in signs given by the Chevalley pairing. Since ? = −em · · · · e2 · e1, we havethat
?2 = (−1)m(m−1)
2
and hence it splits the space of forms into its eigenspaces, namely, into its ±1-eigenspaces∧•±V ∗ if m is zero or one modulo four or its ±i-eigenspaces if n is 2 or 3 modulo 4. Thisallows us to define self-dual and anti self-dual forms in dimensions zero and one modulofour. However we would like to have a single definition for all dimensions, so we introducethe concept of SD- and ASD-forms:
Definition 1.4. We say that a form ϕ ∈ ∧•V ∗C is SD if ?ϕ = −im(m−1)
2 ϕ and that it is ASD
if ?ϕ = im(m−1)
2 ϕ. We denote the space of SD-forms by ∧•+V ∗ and the space of ASD-formsby ∧•−V ∗.1
The Clifford action of elements in V± either preserves or switches the eigenspaces of ?:
Lemma 1.5. Let v± ∈ V±. Then acting via Clifford action on forms we have
v+? = (−1)m−1 ? v+ and v−? = (−1)m ? v−
hence for m even,
V+ : ∧•±V ∗ −→ ∧•∓V ∗ and V− : ∧•±V ∗ −→ ∧•±V ∗
1The choice of signs on the Chevalley pairing and of ? were made so that on a four manifold, the notion ofSD and ASD agrees with the usual notions of self-dual and anti self-dual on 2-forms.
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and for m odd
V+ : ∧•±V ∗ −→ ∧•±V ∗ and V− : ∧•±V ∗ −→ ∧•∓V ∗
Proof. For v− ∈ V−, we have that 〈v−, V+〉 = 0 hence v− graded commutes with ?. Forv+ ∈ V+, we can choose an orthonormal basis for V+ for which e1 = v/‖v‖, so that v anticommutes with all the remaining elements of the basis and commutes with e1.
For the rest of this section we will introduce structures on V which force its dimensionto be even so we let m = 2n.
Definition 1.6. A generalized complex structure on V is a complex structure on V whichis orthogonal with respect to the natural pairing. A generalized Hermitian structure or aU(n)×U(n) structure on V is a generalized complex structure J 1 on V and a generalizedmetric G such that J 1 and G commute.
Given a generalized Hermitian structure (J 1,G) on V , J 2 = GJ 1 is orthogonal withrespect to the natural pairing and squares to −Id, hence it is also a generalized complexstructure. Since πV : V± −→ V are isomorphisms, and J 1|V± is a complex structure onV± orthogonal with respect to the natural pairing, it induces complex structures I± on Vcompatible with the metric g induced by G making V into a bi-Hermitian vector space.We can further form the corresponding Hermitian forms ω± = g I±.
Given any generalized Hermitian structure, ∧•V ∗C splits as the intersections of theeigenspaces of J 1 and J 2: Up,q = UpJ 1
∩ U qJ 2, where UpJ i is the ip-eigenspace of J i in
∧•V ∗C . In this context, the generalized Hodge star is related to the action of Ji = eπJ i2 ,
namely:
Lemma 1.7. (Gualtieri [19]) In a generalized Hermitian vector space one has
? = −J1J2.
This means that we can read the decomposition of forms into SD and ASD from theUp,q decomposition, namely ?|Up,q = ip+q. If we plot the (nontrivial) spaces Up,q in alattice, each diagonal is made either of SD- or ASD-forms, with Un,0 made of SD-forms:
Definition 1.8. A positive U(n) structure or a positive Hermitian structure on V is ageneralized metric G and a complex structure I+ on V+, the +1-eigenspace of G, orthogonalwith respect to the natural pairing and a negative U(n) structure or negative Hermitianstructure on V is a generalized metric with an orthogonal complex structure I− on its−1-eigenspace. We say that a generalized complex structure J extends a positive/negativeU(n) structure (G, I) if I is the restriction of J to the appropriate space and (G,J ) is ageneralized Hermitian structure.
Given a generalized Hermitian structure, since J 1 and G commute, J 1 preserves theeigenspaces of G and hence, upon restriction to V±, one obtains a positive and a negativeHermitian structure. Conversely, a positive (resp. negative) Hermitian structure can beextended to V by declaring that it vanishes on V− (resp. V+). Then a pair of positive and
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SD
%%ASD
%%
U0,2
SD
##
U−1,1 U1,1
U−2,0 U0,0 U2,0
U−1,−1 U1,−1
U0,−2
(1.2)
Figure 1: Representation of the spaces of SD and ASD in terms of the (p, q)-decomposition offorms on a 4-dimensional generalized Hermitian structure.
negative Hermitian structures, I+, I− gives rise to a generalized Hermitian structure bydeclaring that J 1 = I+ + I−.
Given a positive U(n) structure on V , we can use the isomorphism V+∼= V to transport
the metric and the complex structure I+ from V+ to V , making it into a Hermitian vectorspace (V, g, I). Further, we can use I+ to define a complex structure I− on V− using theisomorphisms V+
∼= V ∼= V− and this way we have an extension of the U(n) structure to ageneralized Hermitian structure: namely we declare that J 1 is I+ on V+ and I− on V−,hence, in the metric splitting of V, J 1 is the generalized complex structure associated to thecomplex structure2 I and consequenty J 2 is the generalized complex structure associatedto the Hermitian form ω = g I:
J 1 =
(I 00 −I∗
)J 2 =
(0 −ω−1
ω 0
). (1.3)
For this set of choices, there is a relation between the (p, q)-decomposition of formsdetermined by the generalized Hermitian structure and the usual (p, q)-decomposition offorms determined by the complex structure I on V .
2Note that usually, the generalized complex structure associated to a given a complex structure I on a vectorspace differs from J 1 by a sign, or said another way, by overall conjugation. Hence, all the usual concepts getconjugated with this definition, for example, later we will have ∂J 1
= ∂ and the generalized canonical bundle is∧0,nT ∗M
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Proposition 1.9. (Cavalcanti [6]) For J 1 and J 2 as above, the transformation
ψ : V −→ V ψ(X + ξ) = eiω∗ ei2ω−1
∗ (X + ξ) (1.4)
preserves J 1 and maps V ⊂ V into the +i-eigenspace of J 2 and V ∗ ⊂ V into the−i-eigenspace of J 2. Therefore Ψ, the corresponding action on spinors, preserves theeigenspaces of J 1 and maps ∧kV ∗ into Un−kJ 2
that is
Ψ : ∧•V ∗C −→ ∧•V ∗C Ψ(ϕ) = eiωei2ω−1
(ϕ) (1.5)
Ψ(∧p,qV ∗) = U q−p,n−p−qJ 1,J 2.
Proof. Since this statement is slightly different from that found in [6], we give a shortsketch of the proof. Firstly, a simple computation gives an explicit expression for ψ:
ψ(X + ξ) = X − i
2ω−1(ξ) +
ξ
2− iω(X).
Then the first claim is proved by restricting ψ to V and to V ∗. Indeed, if we let L2
denote the +i-eigenspace of J 2 then
ψ(X) = X − iω(X) ∈ L2 ∀X ∈ VC;
ψ(ξ) =1
2(ξ − iω−1(ξ)) ∈ L2 ∀ξ ∈ V ∗C
(1.6)
Since ω is of type (1, 1) it maps V 0,1 into V ∗0,1 hence it preserves the ±i-eigenspaces ofJ 1.
Finally, since the decomposition of forms associated to the bialgebroid V, V ∗ ⊂ V isgiven by the usual degree of forms, we see that the map above maps degree k forms intothe space Un−k for the symplectic structure.
A positive Hermitian structure is fully determined by V 1,0+ , the +i-eigenspace of I+,
that is, a positive U(n) structure is a subspace V 1,0+ ⊂ TCM for which
• dim(V 1,0) = n;
• V 1,0+ is isotropic with respect to the natural pairing;
• V 1,0+ ∩ V 1,0
+ = 0 and
• 〈v, v〉 > 0 for all v ∈ V 1,0+ \0.
If we are given a generalized complex extension J of a positive Hermitian structure(G, I+) we obtain a bigrading of forms into Up,q as explained earlier. However, from thepoint of view of the U(n) structure, the natural spaces to consider are W k
+ =∑
p+q=k Up,q.
Indeed Wn+ corresponds to the vector space of all forms that are annihilated by the Clifford
action of V 1,0+ and Wn−2k
+ = ∧kV 0,1+ ·Wn
+, that is, the spaces W k+ are solely determined
by the complex structure I+. It is worth calling attention to the fact that W k+ is only
nontrivial if −n ≤ k ≤ n and n − k = 0 mod 2. Further, since the spaces W k+ are the
diagonals of the Up,q decomposition, each W k+ is made of either SD- or ASD-forms, with
Wn+ being SD.
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Another description of the spaces W •+ is obtained by extending I+ to an endomorphismof TM obtained by declaring that I+|V− vanishes, so that I+ is a skew-symmetric operatoron TM with respect to the natural pairing, that is, I+ ∈ spin(TM). Similarly to ageneralized complex structure, J , for which the space Uk is the ik-eigenspace of the actionof J , letting I+ act on forms one sees that the space W k
+ is the ik2 -eigenspace of I+. Hence,Lemma 1.7 takes the following form for positive U(n) structures:
Lemma 1.10. Let (G, I+) be a positive Hermitian structure on V , and let I+ = eπI+2 .
ThenI2+ = −?,
that is ?|Wk+
= −ik.
Similarly, for a negative Hermitian structure (G, I−), we can extend I− to V by declar-ing that it vanishes on V+. If J 1 is a generalized complex extension of I−, then theeigenspaces of I− correspond to the anti-diagonals of the Up,q decomposition: Wk
− =∑p−q=k U
p,q.
Finally we observe that the spaces W k± have both even and odd forms, so one can refine
this grading to a Z× Z2-grading:
W k,0± = W k
± ∩ (∧evV ∗ ⊗ C), W k,1± = W k
± ∩ (∧odV ∗ ⊗ C). (1.7)
In what follows we will refer to both spaces W k± and W k,l
± , with the understanding thatif the Z2-grading is not particularly important, we will simply omit it.
2 Intrinsic torsion of generalized Hermitian struc-
tures
Except for a generalized metric, each of the structures introduced in Section 1 has anappropriate integrability condition. We let (M2n, H) be a manifold endowed with a realclosed 3-form H and consider the Courant bracket on sections of TM = TM ⊕ T ∗M :
[[X + ξ, Y + η]]H = [X,Y ] + LXη − iY dξ − iY iXH.
We will omit the 3-form from the bracket if it is clear from the context.The Courant bracket is the derived bracket associated to the operator dH , i.e., the
following identity holds for all v1, v2 ∈ Γ(TM) and ϕ ∈ Ω•(M):
[[v1, v2]]H · ϕ = v1, dH, v2ϕ
= dH(v1 · v2 · ϕ) + v1 · dH(v2 · ϕ)− v2 · dH(v1 · ϕ)− v2 · v1 · dHϕ,(2.1)
where · denotes the Clifford action of Clif(TM) on ∧•T ∗M and ·, · denotes the gradedcommutator of operators. We will use general properties of the derived bracket throughoutthese notes, and refer the reader to [25] for a concise presentation of its main properties.
The orthogonal action of a 2-formB ∈ Ω2(M) on TM relates different Courant brackets:
[[eB∗ v1, eB∗ v2]]H = eB∗ [[v1, v2]]H+dB.
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Definition 2.1. For each of the structures introduced in Definitions 1.6 and 1.8, we referto the smooth assignment of such structure to TxM for each x ∈ M by including theadjective almost in the name of the structure.
Definition 2.2 (Integrability conditions).
• A generalized complex structure is a generalized almost complex structure J whose+i-eigenspace is involutive with respect to the Courant bracket.
• A generalized Hermitian structure is a pair (G,J 1) of generalized metric and com-patible integrable generalized complex structure.
• A positive (resp. negative) SKT structure is a positive (resp. negative) almost U(n)structure (G, I) for which the +i-eigenspace of I is involutive.
• A generalized (almost) complex structure J extends an SKT structure if J is fiber-wise an extension of I, in which case we say that J is a generalized (almost) com-plex/Hermitian extension of the SKT structure.
• A generalized Kahler structure is a generalized Hermitian structure (G,J 1) for whichJ 1 and J 2 = J 1G are integrable.
2.1 The Nijenhuis tensor
Let us spend some time to understand the Nijenhuis tensor of an almost generalized com-plex structure J . This tensor is defined in the usual way, namely if L is the +i-eigenspaceof J
Nij : Γ(L)× Γ(L) −→ Γ(L); Nij(X,Y ) = −[[X,Y ]]L, (2.2)
where ·L indicates projection onto L. We can alternatively use the natural pairing to define
N : Γ(L)× Γ(L)× Γ(L) −→ Ω0(M ;C);
N(X,Y, Z) = −〈[[X,Y ]], Z〉 = 〈Nij(X,Y ), Z〉.(2.3)
As usual, Nij is a tensor, indeed, for f ∈ C∞(M ;C) we have
Nij(X, fY ) = −[[X, fY ]]L = −(f [[X,Y ]] + (LπT (X)f)Y )L = −f [[X,Y ]]L = fNij(X,Y ).
Further, the tensor N ∈ ∧2L⊗L defined above actually lies in ∧3L. Indeed, for X,Y, Z ∈Γ(L) we have
0 = LπT (X)〈Y,Z〉 = −(〈[[X,Y ]], Z〉+ 〈Y, [[X,Z]]〉) = N(X,Y, Z) +N(X,Z, Y ),
which shows that N is fully skew.A different way to understand N arises by using the Uk decomposition determined by
J . Namely, letting Uk = Γ(Uk) (throughout the paper we denote the sheaf of sections ofUk by Uk and the sheaf of sections of Up,q by Up,q), one has:
Lemma 2.3. In an generalized almost complex manifold
dH : Uk −→ Uk−3 ⊕ Uk−1 ⊕ Uk+1 ⊕ Uk+3. (2.4)
Further the map πk+3 dH : Uk −→ Uk+3 corresponds to the Clifford action of N , theNijenhuis tensor defined in (2.3).
12
Proof. We prove first that
dH : Uk −→∑j≥k−3
U j (2.5)
and will do so by induction, starting at k = n+1 and working our way down. For k = n+1we have that Un+1 = 0 and the claim follows trivially.
Next we assume the result to be true for all j > k and let ρ ∈ Uk. For v1, v2 ∈ Γ(L),using (2.1) we have
v2 · v1 · dHρ = −[[v1, v2]] · ρ+ dH(v1 · v2 · ρ) + v1dH(v2 · ρ)− v2d
H(v1 · ρ).
Since the Clifford action of vi sends U j to U j+1, the inductive hypothesis implies that thelast three terms lie in ⊕j≥k−1U j , while the first term, being the action of an element ofL⊕ L on ρ lies in Uk+1 ⊕ Uk−1. Therefore we conclude that
v2 · v1 · dHρ ∈ ⊕j≥k−1U j for all v1, v2 ∈ Γ(L)
and (2.5) follows.
As dH is a real operator and U−k = Uk conjugating (2.5) we have that
d : Uk −→∑j≤k−3
U j .
Furthermore, if Uk is made of even forms then Uk+1 is made of odd forms and vice versa,we have that
dH : Uk −→∑
−3≤j−k≤3
j−k=1 mod 2
U j ,
therefore proving (2.4).
Now we prove that πk+3 dH corresponds to the Clifford action of N . Once again weuse induction, this time starting at k = −n− 1 and moving upwards. Since U−n−1 = 0,the claim is trivial there. Assume now that for j < k we have proved that πj+3 dH is theClifford action of N . Let ρ ∈ Uk and v1, v2 ∈ Γ(L). Then using (2.1) we have
v2v1πk+3 dHρ = πk+1(v2v1dHρ)
= πk+1(−[[v1, v2]]ρ+ dHv1v2ρ+ v1dHv2ρ− v2d
Hv1ρ)
= N(v1, v2)ρ+ πk+1 dH(v1v2ρ) + v1πk+2 dHv2ρ− v2πk+2 dHv1ρ
= N(v1, v2)ρ+Nv1v2ρ+ v1Nv2ρ− v2Nv1ρ
= N(v1, v2)ρ+N(v1)v2ρ− v2N(v1)ρ+ v2v1Nρ
= N(v1, v2)ρ+N(v1)v2ρ−N(v1, v2)ρ−N(v1)v2ρ+ v2v1Nρ
= v2v1Nρ,
where in third equality we have used that the component of [[v1, v2]]ρ in Uk+1 is givenby the Clifford action of the L component of [[v1, v2]], in the fourth equality we used theinductive hypothesis and in the rest we used basic properties of the Clifford algebra.
13
If we compose dH |Uk with projection onto Uk+1 and Uk−1 we get operators
∂J = πk+1 dH : Uk −→ Uk+1 and ∂J = πk−1 dH : Uk −→ Uk−1,
and if the generalized complex structure J is clear from the context, we denote theseoperators simply by ∂ and ∂. As proved by Gualtieri in [21], integrability is equivalent tothe requirement
dH : Uk −→ Uk−1 ⊕ Uk+1; dH = ∂ + ∂,
so we can also see from this point of view how the vanishing of the Nijenhuis tensor impliesintegrability.
Since (dH)2 = 0, the operators ∂, ∂, N , and N satisfy some relations.
Corollary 2.4. In an generalized almost complex manifold, the following relations andtheir complex conjugate hold
N2 = 0, N, ∂ = 0, N, ∂+ ∂2 = 0, ∂, ∂+ N,N = 0,
where a, b = ab+(−1)|a||b|+1ba is the graded commutator of the operators a and b, which,since all the operators above are odd, becomes simply ab+ ba.
2.2 The intrinsic torsion and the road to integrability
With this understanding of the Nijenhuis tensor, we can give a pictorial description ofthe long road to integrability from almost generalized Hermitian to generalized Kahler.Indeed, given an almost generalized Hermitian structure, we get a splitting of forms intospaces Up,q. According to Lemma 2.3, dH can not change either the ‘p’ or the ‘q’ gradingby more than three and it must switch parity. Hence dH decomposes as a sum of eightoperators and their complex conjugates
dH = δ+ + δ+ + δ−+ δ−+N+ +N+ +N−+N−+N1 +N1 +N2 +N2 +N3 +N3 +N4 +N4;
δ+ : Up,q −→ Up+1,q+1, δ− : Up,q −→ Up+1,q−1,
N+ : Up,q −→ Up+3,q+3, N− : Up,q −→ Up+3,q−3,
N1 : Up,q −→ Up−1,q+3, N2 : Up,q −→ Up+1,q+3,
N3 : Up,q −→ Up+3,q+1, N4 : Up,q −→ Up+3,q−1.
(2.6)
and we can draw in a diagram all the possible nontrivial components of dH |Up,q as arrows(see Figure 2).
Once we require that J 1 is integrable, i.e. we are dealing in fact with a generalizedHermitian structure, then dH only changes the ‘p’ degree by ±1 and several componentsof dH present in the nonintegrable case, now vanish and the diagram from Figure 2 clearsup to the one presented in Figure 3.
Finally, if we require that J 2 is also integrable, and hence we are in fact dealing witha generalized Kahler structure, the last two components of the Nijenhuis tensor, labeledN1 and N2 above, vanish and dH decomposes as a sum of four operators, as pictured inFigure 4.
14
Up−3,q+3 Up−1,q+3 Up+1,q+3 Up+3,q+3
Up−3,q+1 Up−1,q+1 Up+1,q+1 Up+3,q+1
Up,q
N−
bb
N1
^^
N4
jjδ−
bb
N3
tt δ+||
N+||
N2
N1
N−
""
δ− ""
N4
**
δ+<<
N3
44N2
@@
N+
<<
Up−3,q−1 Up−1,q−1 Up+1,q−1 Up+3,q−1
Up−3,q−3 Up−1,q−3 Up+1,q−3 Up+3,q−3
Figure 2: Representation of the nontrivial components of dH when restricted to Up,q for angeneralized almost Hermitian structure.
Up−1,q+3 Up+1,q+3
Up−1,q+1 Up+1,q+1
Up,q
N1
]]
δ−
ff
δ+xx
N2 N1
δ−&&
δ+
88
N2
AA
Up−1,q−1 Up+1,q−1
Up−1,q−3 Up+1,q−3
Figure 3: Representation of the nontrivial components of dH for a generalized Hermitian structure.
This shows that the obstruction for a generalized almost Hermitian structure to bea generalized Kahler structure is given by the tensors Ni, i = 1, 2, 3, 4 and N±, that is,these tensors are the different components of the intrinsic torsion of a generalized almostHermitian structure.
Definition 2.5. The intrinsic torsion of a generalized Hermitian manifold are the tensorsNi, i = 1, 2, 3, 4 and N±.
15
Up−1,q+1 Up+1,q+1
Up,qδ−
cc
δ+
δ−
##
δ+
;;
Up−1,q−1 Up+1,q−1
Figure 4: Representation of the nontrivial components of dH for a generalized Kahler structure.
3 SKT structures
In this section we provide properties equivalent to integrablity of an SKT structure as de-fined in 2.2. We will see that an SKT structure lies precisely half way between a generalizedHermitian and a generalized Kahler structure.
Firstly, we observe that the SKT condition can also be phrased in terms of the vanishingof components of the intrinsic torsion. Indeed, in the presence of an extension of, say, apositive U(n) structure (G, I+) to a generalized almost complex structure J 1 if we let V 1,0
+
be the +i-eigenspace of I+ then involutivity is equivalent to
〈[[v1, v2]], w1〉 = 0 for all v1, v2 ∈ Γ(V 1,0+ ), w ∈ Γ(V 1,0
+ ⊕ V−),
that is, the components N2, N3 and N+ of the intrinsic torsion vanish. A similar argumentgives a characterization of negative SKT structures.
Proposition 3.1. Let G be a generalized metric on a manifold with 3-form (M,H) andlet I± be a pair of positive and negative Hermitian structures compatible with G. If J 1 =I+ +I− then (G, I+) is an SKT structure if and only if N2, N3 and N+ vanish and (G, I−)is a negative SKT structure if and only if N1, N4 and N− vanish.
Corollary 3.2 (Gualtieri [20]). Let G be a generalized metric on a manifold with 3-form(M,H), I± be a pair of positive and negative SKT structures compatible with G andJ 1 = I+ + I−. Then (G,J 1) is a generalized Kahler structure.
Proof. According to Proposition 3.1, under the hypothesis, all components of the intrinsictorsion vanish hence (G,J 1) is a generalized Kahler structure.
Next, we describe the integrability condition for an SKT structure in terms of thedecomposition of forms into W k
± and Up,q (for a fixed generalized complex extension J 1)described in the previous section. Throughout the paper we denote the sheaf of sectionsof W k
± by Wk± and the sheaf of sections of W k,l
± by Wk,l± .
Theorem 3.3. Let (G, I+) be a positive almost Hermitian structure on a manifold with3-form (M2n, H). The the following are equivalent:
1. (G, I+) is a positive SKT structure;
16
Up−3,q+3 Up−1,q+3
Up−3,q+1 Up−1,q+1 Up+1,q+1
Up,q
N−
bb
N1
^^
N4
jjδ−
bb
δ+||
N1
N−
""
δ− ""
N4
**
δ+<<
Up−1,q−1 Up+1,q−1 Up+3,q−1
Up+1,q−3 Up+3,q−3
Figure 5: Representation of the nontrivial components of dH for a positive SKT structure
Up+1,q+3 Up+3,q+3
Up−1,q+1 Up+1,q+1 Up+3,q+1
Up,q
δ−bb
N3
tt δ+||
N+||
N2
δ− ""
δ+<<
N3
44N2
@@
N+
<<
Up−3,q−1 Up−1,q−1 Up+1,q−1
Up−3,q−3 Up−1,q−3
Figure 6: Representation of the nontrivial components of dH for a negative SKT structure
2. dH :Wk+ −→Wk−2
+ ⊕Wk+ ⊕Wk+2
+ for all k;
3. dH :Wn+ −→Wn−2
+ ⊕Wn+.
Proof. One can give a direct proof of this result using only the positive almost Hermitiandata with an argument completely analogous to that already given in Lemma 2.3. Yet,since we have already developed the tools for generalized almost Hermitian structures, wefollow a less intrinsic route.
We will first prove that 1) implies 2). Let I− be any complex structure on V− orthogonal
17
with respect to the natural pairing so that G and J 1 = I+ +I− form a generalized almostHermitian structure. Then according to Proposition 3.1, N2, N3 and N4 vanish. Then, wesee that dH splits into three components:
dH = δN+ + δN+ + /δ− (3.1)
/δ− :Wk+ −→Wk
+; /δ−= δ− + δ− +N− +N−,
δN+ :Wk+ −→Wk+2
+ ; δN+ = δ+ +N1 +N4,
δN+ :Wk+ −→Wk−2
+ ; δN+ = δ+ +N1 +N4.
Condition 2) clearly implies condition 3). Finally to prove that 3) implies 1) we onceagain choose a complex structure I− on V− and observe that since N2, N3 and N+ aretensors, it is enough to check that they vanish when applied to spaces where their actionis effective. But for ϕ ∈ Un,0 ⊂ Wn
+, the components of dH landing on Wn−6+ and Wn−4
+
are precisely N+ · ϕ and N3 · ϕ and the vanishing of these forms for ϕ 6= 0 implies thatN+ = N3 = 0. Similarly, if ψ ∈ U0,n, the Wn−4
+ component of dH is N2 · ψ, hence thevanishing of this component implies that N2 = 0 and Proposition 3.1 implies that I+ isintegrable.
While the different Up,q components of dH obtained in terms of the generalized almostHermitian extension of the SKT structure depend on the particular extension chosen, theoperators δN+ , δN+ and /δ− depend only on the SKT data, since they are the decompositionof dH obtained from the eigenspaces of I+.
Corollary 3.4. The following hold
(δN+ )2 = (δN+ )2 = 0; δN+ , /δ− = 0; δN+ , /δ− = 0; δN+ , δN+ + (/δH− )2 = 0;
where ·, · indicates the graded commutator of the operators.
In the arguments up to now, given a positive SKT structure, we have chosen thecomplex structure I− rather freely, but one can still require some integrability of I−without restricting the geometry, as we argue next.
Proposition 3.5. (Cavalcanti [7]) Given a positive SKT structure (G, I+) on (M,H),there is a generalized complex structure J 1 of complex type which extends I+. FurtherJ 2 = GJ 1 is of symplectic type and its canonical bundle has nonvanishing section ρ2 suchthat ∂J 1ρ2 = 0.
Idea of the proof. Indeed, we can use the procedure from Section 1 to produce an almostcomplex structure I with compatible metric g on M and we can extend the SKT structureto an almost Hermitian structure (G,J 1), where J 1 is the generalized complex structurecorresponding to the complex structure I, as in (1.3). We let ω = gI be the correspondingHermitian form. Then integrability of the SKT structure is equivalent to the integrabilityof I (and hence of J 1) and the condition dcω = H, where H is the 3-form associated to themetric splitting of TM . In particular J 1 is a generalized complex extension of the SKTstructure. Finally, the canonical bundle of J 2 is generated by ρ2 = eiω and
dHeiω = eiω ∧ (H + idω) = 2ieiω∂ω ∈ U1J 1,
hence ∂J 1ρ2 = 0.
18
Theorem 3.6. Let (M,H) be a manifold with 3-form and (G,J 1) be an generalized almostHermitian structure on M . Then the following are equivalent:
1. (G,J 1) is an integrable generalized complex extension of a positive SKT structure;
2. dH : Up,q −→ Up+1,q+1⊕Up+1,q−1⊕Up+1,q−3⊕Up−1,q+3⊕Up−1,q+1⊕Up−1,q−1 for allp, q ∈ Z;
3. dH : Up,n−p −→ Up+1,n−p−1 ⊕ Up+1,n−p−3 ⊕ Up−1,n−p+1 ⊕ Up−1,n−p−1 for all p ∈ Z;
4. dH : U0,n −→ U−1,n−1 ⊕ U1,n−1 ⊕ U1,n−3 and dH : Un,0 −→ Un−1,1 ⊕ Un−1,−1.
Proof. The fact that 1) implies 2) follows from Proposition 3.1 and integrability of J 1.The implications 2)⇒ 3)⇒ 4) are immediate. Finally, similarly to the proof of Theorem3.3, the values of the different components of the intrinsic torsion are fully determined bytheir action on Un,0 and U0,n and 4) implies the vanishing of all components of the intrinsictorsion except from N1.
In light of Proposition 3.5, this theorem allows us to define six differential operators ona generalized complex extension of a positive SKT structure:
dH = δ− + δ− +N +N + δ+ + δ+ (3.2)
δ− : Up,q −→ Up+1,q−1; δ+ : Up,q −→ Up+1,q+1; N : Up,q −→ Up−1,q+3.
Up−1,q+3
Up,q+2
Up−1,q+1 Up+1,q+1
Up,q
N
δ+vv
δ−
hh
δ−((
δ+
66
N
[[
Up−1,q−1 Up+1,q−1
Up,q−2
Up+1,q−3
Figure 7: Decomposition of dH for a generalized complex extension of a positive SKT structure.
We see that N = N1 is the Nijenhuis tensor of J 2, in particular, N is tensorial and thistheorem shows that if (G,J 1) is a generalized Hermitian extension of an SKT structure
19
then half of the Nijenhuis tensor of J 2 vanishes, namely the component N2 from (2.6).Also, this decomposition allows us to express the operators δN+ and /δ− from (3.1) in termsof δ+, δ−, N and their conjugates:
δN+ = δ+ +N ; /δ− = δ− + δ−. (3.3)
Since (dH)2 = 0 these operators satisfy a number of relations.
Corollary 3.7. The following relations and their complex conjugates hold:
δ2+ = N2 = 0; δ+, N = 0;
δ2− = −N, δ+; N, δ− = 0; δ−, δ+ = 0; δ+, δ− = −δ−, N;
δ+, δ++ δ−, δ−+ N,N = 0.
Corollary 3.8. Let (G,J 1) be a generalized Hermitian structure on a manifold with 3-form (M,H) and let J 2 = GJ 1 be the associated generalized almost complex structure. Ifthe canonical bundle of J 2 admits a ∂J 1-closed trivialization, then (G,J ) is an extensionof an SKT structure.
Proof. Since J 1 is integrable, dH : Un,0 −→ Un−1,1⊕Un−1,−1. Due to Lemma 2.3, we havethat
dH : U0,n −→ U−1,n−3 ⊕ U−1,n−1 ⊕ U1,n−1 ⊕ U1,n−3.
But according to the hypothesis there is a trivialization ρ of U0,n such that ∂J 1ρ = 0, thatis, for such ρ the components of dHρ lying in U−1,n−3⊕U−1,n−1 vanish. In particular, thecomponent in U−1,n−3 is part of the Nijenhuis tensor of J 2 and since it vanishes on ρ it isthe zero tensor. So we see that, in fact
dH : U0,n −→ U−1,n−1 ⊕ U1,n−1 ⊕ U1,n−3.
and the last condition of Theorem (3.6) holds.
The same results hold for negative SKT structures. One should bear in mind, howeverthat the relevant spaces for a negative structure are given by W k
− =∑
p−q=k Up,q, that is,
each W k− is an antidiagonal and integrability is equivalent to
dH :Wk− −→Wk+2
− ⊕Wk− ⊕Wk−2
− ,
or, in terms of the Up,q decomposition obtained by choosing a generalized complex exten-sion, dH decomposes as shown in Figure 8.
So an SKT structure (positive or negative) corresponds to a generalized Hermitianstructure in which half of the Nijenhuis tensor of J 2 vanishes.
20
Up+1,q+3
Up−1,q+1 Up+1,q+1
Up,qδ−
gg
δ+ww
N2
δ−''
δ+
77
N2
CC
Up−1,q−1 Up+1,q−1
Up−1,q−3
Figure 8: Decomposition of dH for a negative SKT structure.
4 Parallel Hermitian and bi-Hermitian structures
4.1 Parallelism and the intrinsic torsion
The existence of a relationship between connections with closed, skew symmetric torsionand the Courant bracket was made evident by Hitchin in [22]. Precisely, given a generalizedmetric on TM , we let H be the 3-form corresponding to the metric splitting and g be theinduced metric on M . Also, for X ∈ Γ(TM) we let X± ∈ Γ(V±) be unique lifts of X toV±, we let π± : TM −→ V± be the orthogonal projections onto V± and πT : TM −→ TMbe the natural projection.
Proposition 4.1 (Hitchin [22]). Let ∇± be the unique metric connection whose torsion isskew symmetric and equal to ∓H. Then
∇+XY = πT π+[[X−, Y+]],
∇−XY = πT π−[[X+, Y−]].
From this proposition, we see that the isomorphisms πT : V± −→ TM relate theconnections with torsion ∓H to the operators
∇+: Γ(V−)× Γ(V+) −→ Γ(V+); ∇+v w= π+[[v, w]], v ∈ Γ(V+), w ∈ Γ(V−), (4.1)
∇−: Γ(V+)× Γ(V−) −→ Γ(V−); ∇−wv = π−[[w, v]], v ∈ Γ(V+), w ∈ Γ(V−). (4.2)
As we will work with the spaces V± directly, we will use ∇± instead of the connectionsthey induce on TM , with the understanding that these are equivalent operators, so, forexample, if ∇+ has holonomy in U(n), then M has an almost Hermitian structure (g, I)which is parallel with respect to ∇+ and via the isomorphism πT : V+ −→ TM , V+ gets apositive almost Hermitian structure which is parallel for ∇+ so that ∇+ has holonomy inU(n) and the converse statement also holds.
21
Definition 4.2. A parallel positive (resp. negative) Hermitian structure is a positive (resp.negative) almost Hermitian structure which is parallel with respect to ∇+, (resp. ∇−).A parallel bi-Hermitian structure or parallel U(n) × U(n)-structure is a triple (g, I+, I−)such that (g, I+) is a parallel positive Hermitian structure and (g, I−) is a parallel negativeHermitian structure.
Proposition 4.3. If the connection ∇+ has holonomy in U(n), the corresponding positivealmost Hermitian structure satisfies
[[Γ(V 1,0+ ),Γ(V 1,0
+ )]] ⊂ Γ(V+ ⊗ C). (4.3)
And conversely, if a positive Hermitian structure satisfies (4.3), then it is parallel withrespect to ∇+.
Similarly, ∇− has holonomy in U(n) if and only if
[[Γ(V 1,0− ),Γ(V 1,0
− )]] ⊂ Γ(V− ⊗ C). (4.4)
Proof. Let I+ be a complex structure on V+ orthogonal with respect to the natural pairingand let V 1,0
+ be its +i-eigenspace. Then, for v1, v2 ∈ Γ(V 1,0+ ) and w ∈ Γ(V−), we have that
〈v1, w〉 = 0, as V+ and V− are orthogonal with respect to the natural pairing. Hence
0 = LπT v1〈w, v2〉= 〈[[v1, w]], v2〉+ 〈w, [[v1, v2]]〉= −〈[[w, v1]], v2〉+ 〈w, [[v1, v2]]〉= −〈∇+
wv1, v2〉+ 〈w, [[v1, v2]]〉.
(4.5)
where in the third equality we used again that V+ and V− are orthogonal with respect tothe natural pairing, and hence [[v1, w]] = −[[w, v1]] and in the fourth equality we used thatv2 ∈ V+ and hence the V− component of [[w, v1]] is annihilated by the natural pairing.
If I+ is parallel with respect to ∇+, then 〈∇+wv1, v2〉 vanishes for all v1, v2 ∈ Γ(V 1,0
+ )and w ∈ Γ(V−) and hence, according to (4.5), so does 〈w, [[v1, v2]]〉, showing that [[v1, v2]]must be orthogonal to V− and hence, a section of V+. Conversely, if , [[v1, v2]] ∈ Γ(V+),then for all w ∈ Γ(V−), 〈w, [[v1, v2]]〉 = 0 hence (4.5) implies that 〈∇+
wv1, v2〉 = 0 showingthat ∇+
wv1 is orthogonal to V 1,0+ . Since V 1,0
+ is a maximal isotropic of V+⊗C, we conclude
that ∇+wv1 ∈ Γ(V 1,0
+ ) for all w ∈ V− and v1 ∈ Γ(V 1,0+ ), hence I+ is parallel with respect to
∇+.
Remark. Here we obtain, in a new light, a well known contrast between connections withtorsion and the Levi–Civita connection regarding integrability. Indeed, for the Levi–Civitaconnection, reduction of the holonomy group to U(n) implies integrability of the complexstructure, but that is known not to be the case for connections with torsion. From ourpoint of view, this is the difference between the reduced holonomy condition
[[Γ(V 1,0+ ),Γ(V 1,0
+ )]] ⊂ Γ(V+)
and the SKT condition[[Γ(V 1,0
+ ),Γ(V 1,0+ )]] ⊂ Γ(V 1,0
+ ).
22
Theorem 4.4. Let (M,H) be a manifold with 3-form, G be a generalized metric on Mand I+, I− be a positive and a negative almost Hermitian structure on M
• I+ is parallel with respect to ∇+ if and only if N2 = N3 = 0, i.e.,
dH :Wk+ −→Wk−6
+ ⊕Wk−2+ ⊕Wk
+ ⊕Wk+2+ ⊕Wk+6
+ ;
• I− is parallel with respect to ∇− if and only if N1 = N4 = 0, i.e.,
dH :Wk− −→Wk−6
− ⊕Wk−2− ⊕Wk
− ⊕Wk+2− ⊕Wk+6
− ;
• (G, I+, I−) is a parallel bi-Hermitian structure if and only if Ni = 0 for i = 1, 2, 3, 4.
Proof. Since V− is the orthogonal complement of V+ with respect to the natural pairing,it is clear that (4.3) is equivalent to the following two conditions
〈[[v1, v2]], w〉 = 0 for all v1, v2 ∈ Γ(V 1,0+ ), w ∈ Γ(V 0,1
− ).
〈[[v1, v2]], w〉 = 0 for all v1, v2 ∈ Γ(V 1,0+ ), w ∈ Γ(V 1,0
− );
but the first condition is equivalent to the vanishing of N2 and the second, to the vanishingof N3. Finally, since the component of dH mapping Wk to Wk+4 is given by the sumN2 + N3, we see that the vanishing of this component is also equivalent to the parallelcondition.
The remaining claims are proved similarly.
The decomposition of dH for an almost generalized Hermitian structure extending aparallel positive Hermitian structure is depicted in Figure 9 and the decomposition of dH
for a parallel bi-Hermitian structure is depicted in Figure 10.
The same argument used in Theorem 4.4, shows that the obstructions for parallelstructures to be integrable are given by N±:
Proposition 4.5. Let (M,H) be a manifold with 3-form.
• A parallel positive (resp. negative) almost Hermitian structure is a positive (resp.negative) SKT structure if and only if N+ = 0 (resp. N− = 0);
• A parallel bi-Hermitian structure is a generalized Kahler structure if and only if N+ =N− = 0.
Corollary 4.6. Let M4 be a four dimensional manifold.
• A parallel positive/negative Hermitian structure is a positive/negative SKT structure;
• A parallel bi-Hermitian structure is a generalized Kahler structure.
Proof. Since M is four dimensional, V 1,0± are two dimensional complex vector spaces, so
∧3V 1,0± = 0 and hence N± ∈ ∧3V 1,0
± are the trivial tensors.
We finish this section with a characterization of parallel structures similar to the dif-ferential geometric characterization of SKT structures.
23
Up−3,q+3 Up−1,q+3 Up+3,q+3
Up−3,q+1 Up−1,q+1 Up+1,q+1
Up,q
N−
bb
N1
^^
N4
jjδ−
bb
δ+||
N+||
N1
N−
""
δ− ""
N4
**
δ+<<
N+
<<
Up−1,q−1 Up+1,q−1 Up+3,q−1
Up−3,q−3 Up+1,q−3 Up+3,q−3
Figure 9: Representation of the nontrivial components of dH for an almost generalized Hermitianstructure extending a parallel positive Hermitian structure
Up−3,q+3 Up+3,q+3
Up−1,q+1 Up+1,q+1
Up,q
N−
cc
δ−bb
δ+||
N+
N−
##
δ− ""
δ+<<
N+
;;
Up−1,q−1 Up+1,q−1
Up−3,q−3 Up+3,q−3
Figure 10: Representation of the nontrivial components of dH for a parallel bi-Hermitian struc-ture.
Theorem 4.7. Let M be a manifold with 3-form, G be a generalized metric and I+, I−be a positive and a negative almost Hermitian structure on M and J 1 = I+ + I−. Let Hbe the 3-form corresponding to the metric splitting of TM and (g, I±) be the correspondingbi-Hermitian data. Then
1. I+ is a parallel positive Hermitian structure if and only if
dω+ = I∗+H;
where ω+(X,Y ) = g(I+X,Y ) and I∗+|Ωp,q = i(p− q);
24
2. (G,J 1) is a parallel negative Hermitian structure if and only if
dω− = −I∗−H;
where ω−(X,Y ) = g(I−X,Y );
3. (G,J 1) is a parallel bi-Hermitian structure if and only if both conditions above hold.
Proof. We will only prove 1) as 2) is completely analogous and 3) follows from 1) and 2).Let v, w ∈ Γ(V 1,0
+ ) and let X = πT (v) and Y = πT (w) ∈ Γ(T 1,0+ M) so that v = X+g(X)
and w = Y + g(Y ), where T 1,0+ M is the +i-eigenbundle of the almost complex structure
I+. Since ω+ = g I+, we have that g(X) = −iω+(X) and hence
[[v, w]]H = [[X − iω+(X), Y − iω+(Y )]]H
= [[eiω+X, eiω+Y ]]H
= eiω+ [[X,Y ]]H+idω+
= [X,Y ]− ιY ιX(H + idω+)− iι[X,Y ]ω+.
Therefore, [[v, w]] is a section of V+ if and only if
g([X,Y ]) = −ιY ιX(H + idω+)− iι[X,Y ]ω+. (4.6)
Splitting [X,Y ] into its (1, 0) and (0, 1)-components, we have [X,Y ] = [X,Y ]1,0 −Nij(X,Y ) = [X,Y ]1,0 − ιY , ιX , N, where Nij and N ∈ Γ(∧3LI+) are the tensors intro-duced in (2.2) and (2.3) for the generalized complex structure associated to I+
J I+ =
(−I+ 0
0 I∗+
)computed with respect to the Courant bracket with zero 3-form. Similarly, dω+ has fourcomponents:
dω+ = N · ω+ + ∂ω+ + ∂ω+ +N · ω+.
Hence, (4.6) is equivalent to
−iι[X,Y ]1,0ω+ − iιY , ιX , Nω+ =− ιY ιX(H + i(N · ω+ + ∂ω+ + ∂ω+ +N · ω+))+
− i(ι[X,Y ]1,0ω+ − ιY , ιX , Nω+).
Since ∂ω+ ∈ Ω1,2(M) and N · ω+ ∈ Ω0,3(M), these terms are annihilated by the interiorproduct by X,Y ∈ Γ(T 1,0
+ M) and we obtain that (4.6) is equivalent to
iιY ιXH = ιY ιX∂ω − 2ιY , ιX , N · ω+ + ιY ιXN · ω+. (4.7)
Equating the (0, 1) components of both sides of (4.7), we get
iιY ιXH2,1 = ιY ιX∂ω+,
henceiH2,1 = ∂ω+. (4.8)
25
To determine the (1, 0) components of both sides of (4.7) we contract the whole equationwith Z ∈ T 1,0
+ M :
iH(X,Y, Z) = 2ιY , ιX , N · ιZω+ + ιZιY ιXN · ω+. (4.9)
And we can expand the last term
ιZιY ιXN · ω+ = −ιY , ιX , NιZω + c.p.
Hence, after taking cyclic permutations of (4.9), we have
3iH(X,Y, Z) = ιZιY ιXN · ω,
which is equivalent to3iH3,0 = N · ω+ (4.10)
and (4.7) is equivalent to (4.8) and (4.10). Since H is real, we obtain that the parallelcondition is equivalent to
I∗+H = 3iH3,0 + iH2,1 − iH2,1 − 3iH3,0 = N · ω+ + ∂ω+ + ∂ω+ +N · ω+ = dω+.
5 Hodge theory
In this section we develop Hodge theory for manifolds with parallel Hermitian, bi-Hermitianand SKT structures . Our main result is that for a parallel positive Hermitian structurethe Laplacian preserves the spaces Wk,l
+ and hence induces a decomposition of the dH -cohomology accordingly. This is in contrast with the fact that for usual manifolds thedH -cohomology has only a Z2-grading. For positive SKT manifolds, not only does theLaplacian preserve the spacesWk,l
+ , but in fact there is an identity between the Laplacians
for the operators dH , δN+ and δN+ . We start with real Hodge theory and some operators ofinterest.
5.1 Differential operators, their adjoints and Laplacians
Given a generalized metric and orientation on a compact manifold Mn, we can form theHodge star operator which gives us a positive definite inner product on forms:
G(ϕ,ψ) =
∫M
(ϕ, ?ψ)Ch for all ϕ,ψ ∈ Ω•(M ;R). (5.1)
Two basic results about dH are:
Lemma 5.1. If Mm is compact∫M
(dHϕ,ψ)Ch = (−1)m∫M
(ϕ, dHψ)Ch. (Integration by parts)
G(dHϕ,ψ) = G(ϕ, (−1)m ?−1 dH ? ψ) (Formal adjoint)
hence the formal adjoint of dH is given by dH∗ = (−1)m(m+1)
2 ? dH?.
26
The operators we will be mostly interested in are
/DH+ = 1
2(dH − (−1)m(m−1)
2 ? dH?) = 12(dH + (−1)m+1dH∗)
/DH− = 1
2(dH + (−1)m(m−1)
2 ? dH?) = 12(dH + (−1)mdH∗).
(5.2)
The operators /DH± relate to the projections of dH onto SD- and ASD- forms
dH± : Ω•(M) −→ Ω•±(M),
dH±ϕ =1
2(dHϕ∓ i−
m(m−1)2 ) ? dHϕ.
Lemma 5.2. Given ϕ ∈ Ω•(M) let ϕ± be its SD and ASD components, then
/DH±ϕ = dH∓ϕ+ + dH±ϕ−.
In particular we see that /DH− preserves the spaces Ω•±(M) while /D
H+ maps them to each
other.
Proof.
/DH±ϕ = 1
2((dH ∓ (−1)m(m−1)
2 ? dH?)(ϕ+ + ϕ−)
= 12((dH ± (−1)
m(m−1)2 i
m(m−1)2 ? dH)ϕ+ + 1
2((dH ∓ (−1)m(m−1)
2 im(m−1)
2 ? dH)ϕ−
= 12((dH ± i3
m(m−1)2 ? dH)ϕ+ + 1
2((dH ∓ i3m(m−1)
2 ? dH)ϕ−
= 12((dH ± i−
m(m−1)2 ? dH)ϕ+ + 1
2((dH ∓ i−m(m−1)
2 ? dH)ϕ−
= dH∓ϕ+ + dH±ϕ−.
We note that since /DH+ swaps Ω•+(M) with Ω•−(M) it is a generalization of the signature
operator. One could use this result to define an H-signature and an H-Euler characteritic
as the indices of /D±. But since /DH+ has the same symbol as d + (−1)m+1d∗ and /D
H− has
the same symbol as d+ (−1)md∗, the indices of these operators agree and hence they arejust the usual signature and Euler characteristic.
Proposition 5.3. Let 4H = dHdH∗ + dH∗dH be the dH-Laplacian. Then
1. ?4H = 4H? and
2. (−1)m+1( /DH+ )2 = (−1)m( /D
H− )2 = 1
44H .
Therefore4H preserves the decomposition of forms into Ω•±(M) and hence the dH-cohomologyof M splits as SD- and ASD-cohomology: HdH (M) = HH+ (M)⊕HH− (M).
Proof. The first claim follows by simply expanding dH∗ = (−1)m(m+1)
2 ? dH?:
?4H = (−1)m(m+1)
2 ? (dH ? dH ?+ ? dH ? dH) = (−1)m(m+1)
2 (?dH ? dH ?+ ?2 dH ? dH)
= (−1)m(m+1)
2 (?dH ? dH ?+dH ? dH?2) = 4H ? .
27
The second is just as straightforward:
( /DH+ )2 = 1
4(dH+(−1)m+1dH∗)2 = 14(dH)2+(−1)m+1(dHdH∗+dH∗dH)+(dH∗)2 = (−1)m+1
4 4H .
5.2 Hodge theory on SKT manifolds
Theorem 5.4. Let (M,H) be a compact manifold with 3-form, let (G, I+, I−) be a gen-eralized metric with a pair of almost Hermitian structures and J 1 = I+ + I−. Then
1. If I+ (resp. I−) is a parallel positive (resp. negative) Hermitian structure, the dH-
Laplacian preserves the spaces Wk,l+ (resp. Wk,l
− ) and hence the dH-cohomology of Minherits a corresponding Z× Z2-grading;
2. If (G,J 1) is a parallel bi-Hermitian structure, the dH-Laplacian preserves the spacesUp,q and hence the dH-cohomology of M inherits a corresponding Z2-grading;
Proof. The claim for positive and negative structures are analogous and together theyimply the last claim, so it is enough to prove the first claim.
If (M,H) has a generalized Hermitian structure (G,J 1), then, since each diagonal ofthe Up,q decomposition lies in either Ω•+(M ;C) or in Ω•−(M ;C), so Lemma 5.2 implies that
/DH+ and /D
H− are decomposed into a sum of operators:
/DH+ = δ+ +N1 +N4 +N+ + δ+ +N1 +N4 +N+;
/DH− = δ− +N2 +N3 +N− + δ− +N2 +N3 +N−.
(5.3)
If I+ is parallel, then, according to Theorem 4.4, N2 = N3 = 0 and from (5.3) we have
that /DH− = δ− + δ− +N− +N−, which clearly preserves the Wk
+ decomposition, therefore
4H = 4( /DH− )2 also preserves the Wk decomposition of forms. Since the Laplacian has
even parity, it also preserves Ωev(M) and Ωod(M), hence the Laplacian preserves the
spaces Wk,l+ .
Corollary 5.5. Let M be a manifold with 3-form.
• An SKT structure on M induces a Z× Z2-grading on the dH -cohomology;
• (Gualtieri [19]) A generalized Kahler structure induces a Z2-grading on the dH -cohomology.
For each of the cases covered in the previous theorem, we denote by H•,•dH
(M) the
corresponding decomposition of the dH -cohomology.
Corollary 5.6. Let (M2n, H) be a compact manifold.
• If (G, I+) is a parallel positive (resp. negative) Hermitian structure for which N+ 6≡ 0(resp. N− 6≡ 0), then H±n,•
dH(M) = 0.
• If (G,J 1) is a parallel U(n) × U(n) structure on M for which N+ 6≡ 0 and N− 6≡ 0,then Hp,q
dH(M) = 0 whenever |p+ q| = n or |p− q| = n.
28
Proof. Once again, the proof for positive and negative Hermitian structures is analogousand together they imply the claim for parallel bi-Hermitian structures, so we prove theclaim only for positive structures. Assume that Hn,•(M) is nontrivial. Then there is aharmonic form ρ ∈ Γ(Wn,•
+ ), in particular, ρ is dH -closed, but the component of dHρ in
Wn−3,• is given by N+ρ and the action of ∧•V 0,1+ on Wn is faithful, hence N+ρ = 0 for
ρ 6= 0 implies N+ = 0, contradicting our initial hypothesis.
Corollary 5.7. (Riemann bilinear relations) In a compact parallel positive Hermitianmanifold
i−k(a, a)Ch > 0 for all a ∈ Hk,ldH
(M)\0.
Proof. Indeed, let α be the harmonic representative for the class a ∈ Hk,ldH
(M)\0. Since
α ∈ Wk,l+ , ?α = ikα and hence ?α = i−kα and
i−k(α, α)Ch = (α, ?α)Ch > 0.
Next we prove that the δN+ Laplacian is a multiple of the dH Laplacian in an SKTmanifold. Firstly, we extend the real inner product on forms (5.1) to complex valued formsby requiring it to be Hermitian. If we let ? denote the operator given by ?ϕ = ?ϕ, we have
G(ϕ,ψ) = (ϕ, ?ψ) for all ϕ,ψ ∈ Ω•(M ;C).
Proposition 5.8. Let M2n be a compact generalized almost Hermitian manifold and letδ denote any of the operators δ+, δ+, δ−, δ−, Nα or Nα, where α = 1, 2, 3, 4,±. For ϕ,ψ ∈Ω•(M ;C) we have ∫
M(δϕ, ψ)Ch =
∫M
(ϕ, δψ)Ch (Integration by parts)
δ∗ = ±δ, (Formal adjoint)
where the sign for the adjoint is positive if δ : Ω±(M ;C) −→ Ω±(M ;C) and negativeotherwise.
Proof. The proof that integration by parts holds is the same for all of these operators, so weconsider only δ+. It is enough to consider the case when ϕ ∈ Up,q and hence δ+ϕ ∈ Up+1,q+1
and it pairs trivially with Uk,l, unless k = −p− 1 and l = −q − 1. Hence we may assumethat ψ ∈ U−p−1,−q−1 and compute∫
M(δ+ϕ,ψ)Ch =
∫M
(dHϕ,ψ)Ch =
∫M
(ϕ, dHψ)Ch =
∫M
(ϕ, δ+ψ)Ch,
where in the first equality we have used that the remaining components of dHϕ do not liein Up+1,q+1, hence they pair trivially with ψ, in the second equality we integrated by partsand then reversed the argument.
29
For the formal adjoint, again taking δ = δ+, ϕ ∈ Up,q and ψ ∈ Up+1,q+1 we compute:
G(δ+ϕ,ψ) =
∫M
(δ+ϕ, ?ψ)Ch = i−p−q−2
∫M
(ϕ, δ+ψ)Ch
= i−p−q−2
∫M
(ϕ, (−1)n? ?δ+ψ)Ch = i−p−q−2(−1)n∫M
(ϕ, ? ? δ+ψ)Ch
= i−2p−2q−2(−1)n∫M
(ϕ, ?δ+ψ)Ch = −G(ϕ, δ+ψ),
where we have used several times that on Up,q ? is multiplication by ip+q and in the lastequality we used that p+ q = n mod 2.
Corollary 5.9. (δN+ )∗ = −δN+ .
Theorem 5.10. In a positive SKT manifold,
4δN+= 4
δN+= 1
44H .
Proof. From /DH+ = δN+ + δN+ and Propositions 5.3 and 5.8 we have
144
H = −( /DH+ )2 = −(δN+ + δN+ )2 = −(δN+ )2 − δN+ δN+ − δN+ δN− − δN+
2
= −δN+ δN+ − δN+ δN+ = δN+ δN∗+ + δN∗+ δN+ = 4δN+
Corollary 5.11. The complex
· · · −→ Wk−2+
δN+−→Wk+
δN+−→Wk+2+
δN+−→ · · ·
is elliptic and hence so is the complex for δ+.
Note that δ−, obtained in the presence of a extension of the SKT structure to a gener-alized Hermitian structure, does not square to zero, hence the chain
· · · −→ Up−1,q−1 δ−−→ Up,q δ−−→ Up+1,q+1 δ−−→ · · · (5.4)
is not a complex and its failure to be a complex is measured by δ2− = −δ+, N.
5.3 A spectral sequence
The decomposition δN+ = δ+ + N obtained in the presence of a generalized Hermitianextension of a positive SKT structure is such that δ2
+ = N2 = δ+, N = 0, hence thisdecomposition of the differential gives rise to a spectral sequence converging to the coho-mology of δN+ . The first page of this sequence is just the δ+-cohomology, H•δ+ . The nextnontrivial page corresponds to the cohomology of the operator N : H•δ+ −→ H•δ+ and thesequence goes on.
30
We can concretely compute the following pages of the sequence. Indeed, this is similarto the spectral sequence used to compute the cohomology of dH from the usual de Rhamcohomology [5, 1]. A given ϕp,q ∈ Up,q gives rise to an element in the first page of thespectral sequence if it represents a nontrivial δ+-cohomology class. Then [ϕp,q] survivesthe next step if [ϕp,q] is N -closed, that is N · [ϕp,q] = 0, i.e., if there is ϕp−2,q+2 ∈ Up−2,q+2
such thatN · ϕp,q = δ+ϕ
p−2,q+2 (5.5)
and it bounds if [ϕp,q] is N -exact. If the class survived the second step, it will survive thethird step if for some choice of ϕp−2,q+2 satisfying (5.5), [ϕp−2,q+2] is N -closed and it willbound if it is N -exact, that is, we can find ϕp−4,q+4 ∈ Up−4,q+4 such that
N · ϕp−2,q+2 = −δ+ϕp−4,q+4 (5.6)
Notice that N ·ϕp−2,q+2 is just some kind of Massey product. Indeed, N ·N = N2 = 0 = δ+0and N · ϕp,q = δ+ϕ
p−2,q−2, so if we use the usual expression for Massey products of formswe have
〈N,N,ϕp,q〉 = Nϕp−2,q+2.
So the class of ϕp,q continues to exist after the third step if the Massey product 〈N,N,ϕp,q〉is defined and is the trivial δ+-cohomology class.
The general step of the sequence corresponds to the Massey product 〈k︷ ︸︸ ︷
N,N, · · · , N, ϕp,q〉and the class survived to this stage if there are choices of ϕp−jk,q+2j for j < k, such thatthe previous products are defined and vanish. The class survives this stage if we can chooseϕp−2k,q+2k with
〈k︷ ︸︸ ︷
N,N, · · · , N, ϕp,q〉 = −δ+ϕp−2k,q+2k
and such that the Massey product
〈k︷ ︸︸ ︷
N,N, · · · , N, ϕp,q〉
vanishes (in δ+-cohomology).Since on M2n the bidegree of ϕ is bound by ±n we see that there are at most n + 1
nontrivial steps in the sequence. The element ϕp,q survives in E∞ if all Massey products
〈N,N, · · · , N, ϕp,q〉
vanish in which case the forms ϕj above give rise to a δN+ -closed form:∑n+1
k=0 ϕp−2k,q+2k.
With this argument we have proved
Theorem 5.12. Let (G, I+) be a positive SKT structure on a compact manifold with 3-form (M,H) and let J be a generalized complex structure extending I+. There is a spectralsequence converging to the δN+ -cohomology whose page E2 is the δ+ cohomology and whosedifferentials are Massey products
dkϕ = 〈k−1︷ ︸︸ ︷
N, · · · , N, ϕ〉.
This sequence degenerates after the page En+1.
31
Up−5,q+7
Up−3,q+5
Up−4,q+4
N
aa
δ+ 44
Up−1,q+3
Up−2,q+2
N
__
δ+ 44
Up+1,q+1
Up,qd4
hh
d3
ff
N=d2
\\
δ+=d1
55
Figure 11: Graphic representation of the first four differentials of the spectral sequence relatingδ+- to δN+ -cohomology.
Proposition 5.13. Let Hk,ldH
(M) be the Z × Z2-graded decomposition of dH-cohomology
of a positive SKT manifold induced by the decomposition of forms into the spaces W k,l+ .
Also, let Hp,qδ+
(M) be splitting of the cohomology of the operator δ+ according to the Up,q
decomposition. Then the Euler characteristic and the signature of M are given by
χ =∑
(−1)l dim(Hn−2k,ldH
(M)) = ±∑
(−1)p dim(Hp,qδ+
(M))
σ =∑
(−1)k dim(Hn−2k,ldH
(M)) =∑
(−1)n−p−q dim(Hp,qδ+
(M))
Proof. As we argued earlier, the indices of /DH+ and /D
H− are just σ and χ, signature and
Euler characteristic of M . Then the claims relating H•,•dH
(M) with the χ and σ follow from
the splitting of the dH -cohomology obtained in 5.10.
As for δ+, the proof is an iterated application of the fact that if
0 −→ V0d0−→ V1
d1−→ · · · dn−1−→ Vn −→ 0.
is a complex, then the alternating sum of the dimensions is the same for the complex andits cohomology: ∑
(−1)i dim(Vi) =∑
(−1)i dim
(ker(di)
Im (di−1)
),
and the fact that N maps Ωev(M) −→ Ωod(M) and Ω+(M) into Ω−(M) and vice versa,hence the corresponding alternating sum of the dimensions remains constant for all pagesin the spectral sequence relating the δ+-cohomology with the δN+ -cohomology. The sign tobe used for the Euler characteristic in terms of Hδ+(M) is positive if U0,n is made of evenforms and negative otherwise.
32
5.4 Relation to Dolbeault cohomology
As we saw in Proposition 3.5, given a positive SKT structure (G, I) on M one can extendI+ to a generalized complex structure J 1 of complex type. In this case, in the metricsplitting of TM , the structures J 1 and J 2 = GJ 1 are given by
J 1 =
(I 00 −I∗
), J 2 =
(0 −ω−1
ω 0
),
where ω = g I and for such generalized Hermitian structure Proposition 1.9 gives anautomorphism of the space of forms which relates the Up,q decomposition of forms withthe usual ∧p,qT ∗ decomposition of forms determined by the complex structure I.
In this section we relate these two decomposition of forms and corresponding coho-mologies. Precisely, the effect of applying the automorphism Ψ from (1.5) to ∧•T ∗CM ondH is simply to conjugate it by Ψ, so we get a new operator
dH = Ψ−1 dH Ψ.
Since dH splits according to the Up,q into six operators, the same is true for dH and wecan define
δ+ : Ωp,q −→ Ωp−1,q; δ+ : Ωp,q −→ Ωp+1,q;
δ− : Ωp,q −→ Ωp,q+1; δ− : Ωp,q −→ Ωp,q+1;
N : Ωp,q −→ Ωp−1,q−2 N : Ωp,q −→ Ωp+1,q+2;
dH = δ+ + δ+ + δ− + δ− + N + N .
Proposition 5.14. Let (G, I+) be a positive SKT structure on a manifold with 3-form(M,H), let J be the generalized complex extension of I+ given by Proposition 3.5. Let
∂ω−1
= ∂, ω−1 and ∂ω−1
= ∂, ω−1,
let ζi be the component of ei2ω−1
∗ ∂ω lying in ∧iT ⊗∧3−iT ∗ and let χ be the (1, 2)-componentof [ω−1, ω−1]SN , where [·, ·]SN is the Schouten–Nijenhuis bracket. Then the following hold
δ+ = i2∂
ω−1
+ 2iζ2 δ+ = ∂
δ− = ∂ + 2iζ1 δ− = i2∂
ω−1(5.7)
N = 18χ N = 2i∂ω.
Proof. The proof is a direct computation of the operator dH :
dH = e−i2ω−1
e−iω(d+H∧)eiωei2ω−1
= e−i2ω−1
e−iωeiω(d+ (H + idω)∧)ei2ω−1
= e−i2ω−1
dei2ω−1
+ 2ie−i2ω−1
(∂ω∧)ei2ω−1
(5.8)
33
We compute separately each of the two operators above making up dH :
e−i2ω−1
dei2ω−1
= d+ [d, i2ω−1] + 1
2! [[d,i2ω−1], i2ω
−1] + 13! [[[d,
i2ω−1], i2ω
−1], i2ω−1] + · · ·
The first term is just d = ∂ + ∂, while the second term is a version of the symplecticadjoint of d introduced by Koszul [26], but now obtained in a nonintegrable setting, δω
−1=
d ω−1 − ω−1 d. Since ω is not closed, δω−1
does not square to zero. Integrability of thecomplex structure gives d = ∂ + ∂ and recalling that ω−1 is of type (1, 1) we have
δω−1
= ∂ω−1
+ ∂ω−1
; ∂ω−1
: Ωp,q −→ Ωp,q−1; ∂ω−1
: Ωp,q −→ Ωp−1,q.
Since ω−1 is even, the third term in the series coincides with a multiple of the expressionfor the derived bracket of ω−1 with itself, that is, the Schouten–Nijenhuis bracket:
[[d, i2ω−1], i2ω
−1] = − i2ω−1, d, i2ω
−1 = 14 [ω−1, ω−1]SN .
Since ω−1 is of type (1, 1), integrability of the complex structure implies that this termthis is a 3-vector lies in ∧2,1T ⊕ ∧1,2T , so this terms also decomposes:
[ω−1, ω−1]SN = χ+ χ, χ ∈ ∧1,2TM.
The fourth term is the commutator [[ω−1, ω−1]SN , ω−1] which vanishes since ω−1 is a
bivector and hence so do the remaining terms in the series. So we have established that
e−i2ω−1
de−i2ω−1
= ∂ + ∂ + i2∂
ω−1+ i
2∂ω−1
+ 18(χ+ χ) (5.9)
Next we compute the second summand in (5.8):
2ie−i2ω−1 ∂ω e
i2ω−1
= 2ie−i2ω−1 (e
i2ω−1
∗ e− i
2ω−1
∗ ∂ω) ei2ω−1
= 2ie−i2ω−1 e
i2ω−1 (e
− i2ω−1
∗ ∂ω)
= 2ie− i
2ω−1
∗ ∂ω,
where we have used (1.1) in the second equality. The element e− i
2ω−1
∗ ∂ω ∈ ∧3TCM has sixcomponents:
e− i
2ω−1
∗ ∂ω ∈ (∧2,1T ∗)⊕(T 1,0⊗∧1,1T ∗)⊕(T 0,1⊗∧0,2T ∗)⊕(∧2,0T⊗T ∗1,0)⊕(∧1,1T⊗T ∗0,1)⊕(∧1,2T )
Since, for a 1-form ξ, e− i
2ω−1
∗ ξ = i2ω−1(ξ) + ξ, we see that the 3-form component
is ∂ω and that the 3-vector component is − i8ω−1∂ω. We let ζ1 be the component of
e− i
2ω−1
∗ ∂ω in T ⊗∧2T ∗ and ζ2 be the component in ∧2T ⊗ T ∗, so that ζ1 : ∧p,q −→ ∧p,q+1,ζ2 : ∧p,q −→ ∧p−1,q, and
2ie−i2ω−1 ∂ω e
i2ω−1
= 2i∂ω + 2iζ1 + 2iζ2 +1
4ω−1(∂ω)
Putting this together with (5.9) and the fact that dH does not have a componentmapping Ωp,q(M) to Ωp−2,q−1(M) we conclude that χ = 2ω−1(∂ω) and obtain (5.7).
34
Next, we let ∂i∂ω be the operator ∂ + i∂ω∧. Then we observe that even though ∂i∂ω
does not preserve the degree of a form, it preserves the holomorphic degree, i.e.,
∂i∂ω : Ωp,•(M) −→ Ωp+1,•(M),
so its cohomology has a natural Z-grading. For an operator δ with δ2 = 0 we denote itscohomology by Hδ(M).
Corollary 5.15. Let (G, I+) be a positive SKT structure on a compact manifold, let J bethe generalized complex extension from Proposition 3.5 and let (g, I) be the correspondingHermitian structure with Hermitian form ω = g I. Then
Hp,q∂ (M) ∼= Hq−p,n−p−q
δ+(M),
andHp
∂i∂ω(M) ∼= Hn−2p
δN+(M) ∼= Hn−2p
dH(M).
Proof. Indeed, Ψ puts Ωp,q(M) in correspondence with Uq−p,n−p−q and Ωp,•(M) with
Wn−2p and according to Proposition 5.14, Ψ∂ = δ+ Ψ and Ψ∂2iω = δN+ Ψ. Therefore
the Hp,q∂ (M) ∼= Hq−p,n−p−q
δ+(M) and Hp
∂2i∂ω(M) ∼= Hn−2p
δN+(M). Of course the cohomology
of ∂2i∂ω is the same as the cohomology if ∂i∂ω, as the automorphism
m : ∧•,•T ∗M −→ m : ∧•,•T ∗M m(α) = 2q2α for all α ∈ ∧p,qT ∗M,
relates them.
Interestingly we have that dH = ∂i∂ω + ∂−i∂ω
and the operators ∂i∂ω and ∂−i∂ω
bothsquare to zero (due to the SKT condition ∂∂ω = 0) and graded commute. Yet, they arenot well behaved with respect to the Z2-bigrading of forms and only on a Z2
2-grading dothey behave nicely, as there we have
∂i∂ω : Ωp,q(M) −→ Ωp+1,q(M) ∂−i∂ω
: Ωp,q(M) −→ Ωp,q+1(M), (p, q) ∈ Z22.
There is a standard trick to recover one Z-grading, say the ‘p’-grading. We introduce
a new formal variable, β, alter the operators ∂i∂ω and ∂−i∂ω
so that they act on E1 =Ωp,q(M)⊗ 〈β〉:
∂i∂ω(αβk) = (∂i∂ωα)βk ∂−i∂ω
(αβk) = (∂α)βk − i∂ω ∧ αβk+1
and introduce a Z × Z2-grading on E1 by declaring that for Ωp,q(M) ⊗ βk = Ep−2k,q1 ,
(p− 2k, q) ∈ Z× Z2. I.e., β has degree (−2, 0). With this arrangement
∂i∂ω : Ep,q1 (M) −→ Ep+1,q1 (M) ∂
−i∂ω: Ep,q1 (M) −→ Ep,q+1
1 (M), (p, q) ∈ Z× Z2.
In this setting, one can produce a Z× Z2-graded spectral sequence whose second pageis H•
∂i∂ω(M)⊗ 〈β〉 and the last is H•
dH(M)⊗ 〈β〉. Since these are isomorphic, by Corollary
5.15, this sequence degenerates at the second page.
Theorem 5.16. (SKT Frolicher spectral sequence) In an SKT manifold the decompositiondH = (∂+ i∂ω)+(∂− i∂ω) gives rise to a spectral sequence which degenerates at the secondpage.
35
5.5 Alternative definition
One shortcoming of SKT structures compared the Kahler structures is that there is noanalogue of the dc-operator. Instead, there are two operators, neither of which fully worksas the complex dc. Indeed, given a positive SKT structure (G, I), we can define δI and dI
asδI = dH , I = −i(δN+ − δN+ )
dI = e−πI2 dHe
πI2 = −i(δN+ − δN+ ) + /δ−;
(5.10)
Lemma 5.17. The following hold:
δI∗ = −δI ;
(δI)2 = −144
H δI , dH = 0.
(dI)2 = 0 dI , dH = 124
H ;
(dI∗)2 = 0 dI∗, dH = 1
24H ;
Proof. The claim δI∗ = −δI follows from 5.9. Using Corollary 3.4, Corollary 5.9 andTheorem 5.10 we have
(δI)2 = −(δN+ − δN+ )2 = δN+ δN+ + δN+ δ
N+ = −(δN+ δ
N∗+ + δN∗+ δN+ ) = −1
44H .
Using that the graded commutator satisfies the graded Jacobi identity we have
dH , δI = dH , dH , I = dH , dH, I − dH , dH , I = −dH , δI,
and therefore dH , δI = 0.The conditions (dI)2 = (dI∗)2 = 0 are obvious and for the last equalities we use that
dI = δI + /DH− and dI∗ = −δI + /D
H− , so, for example
dI , dH = δI + /DH− , d
H = /DH− , d
H = /DH− , /D
H−+ /DH
− , /DH+ = 1
24H + 0.
Since (dI)2 = 0, one can also study its cohomology. Not surprisingly, we have thefollowing corollary to Theorem 5.10
Corollary 5.18. Let 4dI be the Laplacian of dI . Then
4dI = 4H .
Proof. Indeed, since dI = δI + /DH− , then dI∗ = −δI + /D
H− and we have
4dI = dI , dI∗
= δI + /DH− ,−δI + /D
H−
= −δI , δI+ /DH− , /D
H−
= 4H ,
where in the fourth equality we used Corollary 3.4, in the fifth we used that δN+ = −δN∗+
and Proposition 5.3 and in the last equality we used Theorem 5.10
36
In light of Lemma 5.17, there seems to be no analogue of the ddc-lemma in the SKTsetting as we do not have enough operators satisfying the correct relations. Yet the operatorδI can be useful in a different context, as we explain next. In [17], Grabowski showed thata generalized complex structure is equivalent to a tensor J ∈ Γ(∧2TM) = Γ(spin(TM))acting on forms via the spin action on spinors and for which the following relation holds
dH ,J ,J = −dH .
One can see where this comes from: for a generalized complex structure we define dJ =dH ,J = −i(∂−∂) and hence dJ ,J = (−i)2(∂+∂) = −dH . So this equation includessimultaneously the fact that J 2 = −Id and the compatibility between J and dH , i.e.,the integrability condition. Further, it is phrased in a way which allows one to work withgeneralized complex structures in the framework of super geometry.
In our setting we can find a similar condition:
Theorem 5.19. Let (M,H) be an oriented manifold with 3-form and G be a generalizedmetric on M . A tensor I ∈ Γ(∧2V+) ⊂ Γ(∧2TM) is an SKT structure if and only if
dH , I, I = − /DH+ , (5.11)
Proof. One implication is easy. If I is an SKT structure. Then dH , I = −i(δN+ − δN+ )
and dH , I, I = (−i)2(δN+ + δN+ ) = − /DH+ .
The converse requires a little more effort. Since I ∈ Γ(∧2TM) = Γ(spin(TM)), it actson forms, thought of as spinors, in a skew symmetric way (with respect to the Chevalleypairing) and hence its eigenvalues are pure imaginary numbers and it splits ∧•T ∗CM into
its eigenspaces, say, W 2λ is the iλ-eigenspace. Since I ∈ Γ(∧2V+), Lemma 1.5 implies that?I = I? and hence I preserves the spaces Ω•±(M) and we can decompose forms into the
intersections of the eigenspaces of I and ?.Let ρ ∈ Γ(W 2λ) be, say, a (locally) nonvanishing SD-form and decompose dHρ into
components:
dHρ =∑µ
ρ+µ + ρ−µ ,
where ρ±µ ∈ Γ(W 2µ) ∩ Ω•±(M). Then
dH , I, Iρ = (I2dH − 2IdH I + dH I
2)ρ
=∑µ
−(µ2 − 2µλ+ λ2)(ρ+µ + ρ−µ )
=∑µ
−(µ− λ)2(ρ+µ + ρ−µ )
On the other hand,/DH+ρ =
∑ρ−µ ,
hence, condition (5.11) implies that /DH−ρ ∈ Γ(W 2λ) and /D
H+ρ ∈ Γ(W 2λ+2 ⊕W 2λ−2) and,
in particular,dH :Wλ −→Wλ+2 ⊕Wλ ⊕Wλ−2 (5.12)
37
If we define define I : TM −→ TM by
Iv = I, v
then I is skew symmetric, hence it decomposes TCM into its eigenspaces and the eigenval-ues of I are all pure imaginary numbers. If v is a iµ-eigenvector of I, then, for ρ ∈W 2λ,
I(v · ρ) = (Iv) · ρ+ v · (Iρ) = (µ+ λ)iv · ρ, (5.13)
that is, the eigenvectors of I map eigenspaces of I to each other. Further since I ∈Γ(∧2V+), if v ∈ V− ⊗ C, Iv = I, v = 0 therefore Clif(V− ⊗ C) preserves the eigenspacesof I.
Since using d, d∗ and multiplication by functions one can generate all forms from any
nonzero form, we conclude that from a nonzero ρ ∈W λ and using /DH− , /D
H+ and the Clifford
action of V− we can generate all forms. Since among those operators /DH+ is the only one
that maps sections of one eigenspace into sections of another and the difference in theeigenvalues is ±1, we see that the eigenvalues of I differ from each other by ik for someinteger k. Since conjugation sends W 2λ to W−2λ, we see that these eigenvalues are eitherall integers or all half integers and the corresponding indexes of W • are either all even or
all odd. Also, since the action of Clif(V− ⊗ C) and /DH− preserve the spaces of SD- and
ASD-forms and the action of /DH+ changes the eigenvalue by ±1 and switches the spaces of
SD and ASD forms, we conclude that for each k, Wk ⊂ Ω•+(M) or Wk ∈ Ω•−(M) and thatif Wk ⊂ Ω•+(M) then W k±2 ∈ Ω•−(M) and vice versa.
The discussion above implies that I|V+ has trivial kernel. Indeed, if v ∈ V+ is in thekernel of I, then for ρ ∈ W k, equation (5.13) implies that v · ρ ∈ W k, but according toLemma 1.5, v · ρ would have reversed SD/ASD status from the elements in W k, hence wewould have v · ρ = 0 for all ρ ∈W k and for all k, hence v must be zero.
Since the eigenvalues of I differ from each other by imaginary integers, equation (5.13)implies that the eigenvalues of I must be imaginary integers. Since I : V+ −→ V+ has nokernel and is skew, we have that half of its eigenvalues (with multiplicity) of the form infor n a positive integer and half are of the form in for n negative. Let v ∈ V+ ⊗ C be anonvanishing section an eigenspace of I, say, Iv = in. Then we can choose f ∈ C∞(M)such that [[v, fv]] = (LπT (v)f)v is nonvanishing. In this case, on the one hand, we have
that for ρ ∈ Wk
[[v, fv]]ρ = (LπT (v)f)v · ρ ∈ Wk+2n.
And on the other hand
[[v, fv]]ρ = v, dH, fvρ ∈ Wk+4n+2 ⊕Wk+4n ⊕Wk+4n−2,
where we have used that v mapsWj intoWj+2n and dH mapsW i intoW i+2⊕W i⊕W i−2. Ifn is not ±1, these spaces do not intersect, and we would get that vρ = 0 for all eigenvectorsρ and hence v · ρ = 0 for all forms, which can not happen for v 6= 0. Hence the eigenvaluesof I are ±i and I2|V+ = −Id. Since I is skew and I2|V+ = −Id, I|V+ is orthogonal, henceit is an almost SKT structure. Then (5.12) implies integrability.
Corollary 5.20. A generalized Kahler structure is a generalized metric G and two tensorsI± ∈ Γ(∧2V±) such that
dH , I±, I± = − /DH± , (5.14)
38
6 Instantons over complex surfaces
Next we use the techniques developed in this section to provide an alternative descriptionof the SKT structure on the moduli space of instantons of a bundle over a complex surface.The tool used to describe this structure are extended actions as introduced in [3] and theSKT reduction theorem as presented in [7]. The argument presented here follows closelythe one from [4], so we will spare details and refer to that paper for further reading.
Let (M, [g], I) be a compact complex surface with a conformal Hermitian structure.By a result of Gauduchon [15], there is a representative g of the conformal class whichmakes (M, g, I) into an SKT manifold, i.e., the corresponding Hermitian form ω satisfiesddcω = 0. We let H = dcω and consider TM endowed with the H-Courant bracket, sothat the metric
G =
(0 g−1
g 0
)and the complex structure on V+ induced by I via the isomorphism π : V+ −→ TM are apositive SKT structure on TM .
Given a bundle E over M with a compact Lie group G as structure group and Liealgebra g we let A be the space of all g-connections on E endowed with the trivial 3-form,so that A is an affine space isomorphic to space of 1-forms on M with values in the adjointbundle gE , Ω1(M ; gE). Hence at any connection D we have TDA = Ω1(M ; gE) and, lettingκ be a bi-invariant metric on G, we can use κ to identify T ∗DA = Ω3(M ; gE). Indeed, forX ∈ Ω1(M ; gE) and ξ ∈ Ω3(M ; gE) we define the natural pairing as
ξ(X) = 2
∫Mκ(X, ξ).
Then for X,Y ∈ Ω1(M ; gE) and ξ, η ∈ Ω3(M ; gE) we have
〈X + ξ, Y + η〉 =
∫Mκ(X, η) + κ(Y, ξ) =
∫Mκ(X, η)− κ(ξ, Y ) =
∫Mκ(X + ξ, Y + η)Ch
where κ(·, ·)Ch indicates that one uses κ to pair elements in gE and the Chevalley pairingon forms to obtain a top degree form.
We denote by G the gauge group of E and by g = Ω0(M ; gE) its Lie algebra. Theinfinitesimal generator corresponding to γ ∈ Ω0(M ; gE) at a point D ∈ A is just the vectorDγ ∈ Ω1(M ; gE) and we can extend this action to form a lifted action as in [7]:
Ψ : Ω0(M ; gE) −→ TA Ψ(γ)|D = DHγ = Dγ +H ∧ γ,
as long as there are no infinitesimal symmetries, i.e., as long as D : Ω0(M ; gE) −→Ω1(M ; gE) has trivial kernel. Therefore, from this point onwards we only consider con-nections for which D : Ω0(M ; gE) −→ Ω1(M ; gE) has trivial kernel. If E is a simpleSU(n)-bundle then that is the case for all ASD connections.
Next we add a moment map to this action. Our (equivariant) moment map takes valueson the G-module h∗ = Ω2
+(M ; gE), the space of self-dual 2-forms:
µ : A −→ Ω2+(M ; gE) µ(D) = (FD)+,
39
where FD is the curvature of the connection D and (·)+ indicates projection onto the spaceof self dual forms, Ω•+(M ; gE), which, for 2-forms in 4 dimensions, agrees with the usualself dual 2-forms. If we let a be the sum g⊕ h, a becomes a Courant algebra if we endowit with the hemisemidirect product:
[[(γ1, λ1), (γ2, λ2)]] = ([γ1, γ2], γ1 · λ2).
Then the maps Ψ and µ together give rise to an extended action given infinitesimally bymap of Courant algebras:
Ψ : a −→ Γ(TA) Ψ(γ, λ)|D = DHγ + d〈µ, λ〉.
We notice that Ω0(M ; gE) + Ω2+(M ; gE) is isomorphic to Ωev
+ (M ; gE) via the map
γ + λ 7→ γ + λ+ ?γ, γ ∈ Ω0(M ; gE), λ ∈ Ω2+(M ; gE),
so we can use a = Ωev+ (M ; gE) and then the extended action is given simply by
Ψ : Ωev+ (M ; gE) −→ Ωod(M ; gE) Ψ(α)|D = DHα.
Following the reduction procedure, the reduced manifold,M, is obtained by taking thequotient of µ−1(0) by the action of the gauge group. In this case, µ−1(0) consists of thespace of anti self-dual connections and hence M is simply the moduli space of instantons.
The reduction procedure also produces a specific Courant algebroid over the reducedmanifold. Namely, if we let K be the bundle generated by Ψ(a) and K⊥ its orthogonalcomplement with respect to the natural pairing, then the space of G-invariant sections ofK⊥/K over µ−1(0) inherits a bracket and a nondegenerate pairing which make the quotientbundle (K⊥/K)/G into a Courant algebroid over M. Notice that at a specific anti self-dual connection D, K is the image of Ωev
+ (M ; gE) by DH . If we let DH+ : Ωod(M ; gE) −→
Ωev+ (M ; gE) be the composition of DH with projection onto the SD-forms, then a simple
integration by parts shows that K⊥|D is the space of DH+ -closed forms and hence the
reduced Courant algebroid is the degree one cohomology of the following elliptic complex:
0 −→ Ωev+ (M ; gE)
DH−→ Ωod(M ; gE)DH+−→ Ωev
+ (M ; gE) −→ 0. (6.1)
In a way, this is the double of the usual elliptic complex describing the tangent spaceto M :
0 −→ Ω0(M ; gE)D−→ Ω1(M ; gE)
D+−→ Ω2+(M ; gE) −→ 0. (6.2)
Earlier we added the assumption that (6.2) above has no cohomology in degree zero. Fromnow on we also add the assumption that this complex has no cohomology in degree two,so that D corresponds to a smooth point inM and hence the dimension ofM is given bythe index of D, which is a topological invariant due to the Atiyah-Singer index theorem.Further, in this case, the cohomology of (6.1) also concentrates in the middle term:
HodDH (M ; gE) =
ker(DH+ : Ωod(M ; gE) −→ Ωev
+ (M ; gE)
Im (DH : Ωev+ (M ; gE) −→ Ωod(M ; gE)
.
40
Since M has an SKT structure, we can also endow A with an SKT structure. Firstlywe let ? be the generalized metric: Indeed, in four dimensions ?2 = Id, spin invariance ofthe Chevalley pairing means that ? is orthogonal (in even dimensions) and by design wehave (c.f. (5.1))
〈ϕ, ?ϕ〉 =
∫Mκ(ϕ, ?ϕ)Ch > 0, for ϕ ∈ Ωod(M ; gE)\0,
so ? is a generalized metric and for this metric V+ is the space of SD odd forms, Ωod+ (M ; gE) =
(W2 +W−2) ∩ Ωod(M ; gE), and V− is the space of ASD odd forms, Ωod− (M ; gE) = W0 ∩
Ωod(M; gE).
For complex structure, we let I = eπI2 as in Lemma 1.10. Then that same lemma
implies that I2|V+ = −? = −Id. Since this structure is independent of the point D ∈ A, it
is constant and hence integrable. The spaces V 1,0+ and V 0,1
+ can be easily described: since
I = I on W±2 we have V 1,0+ =W2 ∩ Ωod(M ; gE) and V 0,1
+ =W−2 ∩ Ωod(M ; gE).Further, since both ? and I are constant and the gauge group acts by translations on
each of its orbits, this SKT structure is invariant by the action of the gauge group. Now,we are in position to use the SKT reduction theorem:
Theorem 6.1 (SKT Reduction Theorem [7]). Let Ψ : a −→ Γ(TA) be an extended actionwith moment map µ preserving an SKT structure (G, I) on A as above and let P = µ−1(0).If the underlying group action on P is free and proper and 0 is a regular value of the momentmap, thenMred = P/G is smooth and the SKT structure on A reduces to an SKT structureon Mred if and only if K⊥ ∩ V+|P is invariant under I.
As we observed earlier, at a point D ∈ A, K⊥ corresponds to the DH+ -closed odd forms
and hence K⊥ ∩ V+ corresponds to the DH+ -closed SD odd forms, that is the SD, DH -
harmonic odd forms and hence according to the theorem to prove that the moduli spaceof instantons inherits an SKT from M , we must prove that the space of SD, DH -harmonicodd forms is invariant under I. To achieve this, one must develop Hodge theory for formswith coefficients. In this case, the condition that the connection is anti self-dual, allows usto achieve the result.
Indeed, similarly to the flat case, we can split DH as a sum of three operators:
DH = δN+ + δN+ + /DH− ,
δN+ :Wk −→Wk+2 δN+ :Wk −→Wk−2 /DH− :Wk −→Wk.
Indeed, locally DH = dH + A, for some A ∈ Ω1(M ; gE) and hence the splitting of dH
together with the splitting of A into its V 1,0+ , V 0,1
+ and V− components gives the desired
decomposition of DH . Just as in Section 5.2, integration by parts gives that (δN+ )∗ = −δN+and /D
H∗− = /D
H− .
Now, let ϕ ∈ (W2 ⊕W−2) ∩ Ωod(M ; gE). Then the DH -Laplacian computed on ϕ isgiven by
4Hϕ = (DH∗DH+ +DHDH∗
+ )ϕ
= (−δN+ − δN+ + /DH− ) /D
H− + (δN+ + δN+ + /D
H− ) /D
H−ϕ
= 2( /DH− )2ϕ.
41
Hence the Laplacian leaves the spacesW2∩Ωod andW−2∩Ωod invariant. Therefore we candecompose the space of harmonic SD odd forms into two spaces H±2 = ker(4H) ∩W±2.I acts as multiplication by i on W2 and by −i on W−2, so either way it preserves theintersection ker(4H) ∩W±2 and hence it preserves the space of SD harmonic odd forms.According to Theorem 6.1, this means that the SKT structure from A reduces to an SKTstructure on M so we have re-obtained the following result, originally due to Lubke andTeleman [28]:
Theorem 6.2. Let (M, I, [g]) be a compact conformal Hermitian 4-manifold, let E be abundle over M whose structure group is compact with Lie algebra g and let gE be theadjoint bundle over M . Let Ms be the quotient of the space
P = D ∈ A : (FD)+ = 0; H0D(M ; gE) = H2
D(M ; gE) = 0,
by the action of the gauge group, i.e., Ms is the smooth locus of the moduli space ofinstantons on E. Then Ms has an SKT structure induced by the unique SKT structureon M in the conformal class [g].
7 Lie algebroid differentials and other operators
Lie algebroids occur naturally in generalized geometry. Notably, if J is a generalizedcomplex structure and L its +i-eigenspace, then L is a Lie algebroid and the correspondingdegree two Lie algebroid cohomology, H2(M ;L∗), governs the infinitesimal deformations ofJ . As we will see next section, similar claims also hold in the SKT case, so it is particularlyrelevant to develop the theory of Lie algebroids related to SKT structures. However, sincesome of the generalized complex structures we consider are not integrable, some bundleswe encounter are not Lie algebroids. Yet some of those “not Lie algebroids” are importantin the study of deformations, so we must develop the relevant part of the theory of thesespaces as well.
7.1 Lie algebroids
In this section we introduce two natural Lie algebroids: one associated to an SKT structureand one to a generalized complex extension of an SKT structure. Then we study therelation between the differential operators defined in the previous section and the naturaloperators defined on these Lie algebroids.
If (M,H) has a positive SKT structure (G, I), by definition V 1,0+ is involutive, hence it
is a Lie algebroid over M . Further V 0,1+ can be identified with the dual of V 1,0
+ using thenatural pairing, which is nondegenerate on V+, hence we get a corresponding differential
∂+ : Γ(∧kV 0,1+ ) −→ Γ(∧k+1V 0,1
+ ). (7.1)
Similarly, given a generalized complex extension J 1 of an SKT structure, L1, the +i-eigenspace of J 1, is also a Lie algebroid and we get a differential on the exterior algebraof the dual dL1 : Γ(∧jL1) −→ Γ(∧j+1L1)
42
Since J 2 = GJ 1 commutes with J 1, it preserves the eigenspaces of J 1 and hencegives rise to a complex structure on L1. In fact, it splits L1 in two subspaces of complexdimension n: V 1,0
+ = L1 ∩ V+ and V 1,0− = L1 ∩ V−, hence ∧•L1 gets a bigrading:
∧k,lL1 = ∧kV 0,1+ ⊗ ∧lV 0,1
− .
Proposition 7.1. Let J 1 be a generalized complex extension of a positive SKT structure.Then
dL1 : Γ(∧k,lL1) −→ Γ(∧k+1,lL1 ⊕ ∧k,l+1L1 ⊕ ∧k−1,l+2L1)
Further,
• The component
N = πk−1,l+2 dL1 : Γ(∧k,lL1) −→ Γ(∧k−1,l+2L1)
corresponds to the Clifford action of the Nijenhuis tensor N ∈ Γ(∧2V 0,1− ⊗ V 1,0
+ )introduced in (3.2);
• The componentπk+1,0 dL1 : Γ(∧k,0L1) −→ Γ(∧k+1,0L1)
corresponds to the operator ∂+ introduced in (7.1).
Proof. Since V 1,0+ is involutive and ∧0,1L1 = V 0,1
− is the space of forms annihilating V 1,0+
in L1, we havedL1 : Γ(∧0,1L1) −→ Γ(∧0,2L1 ⊕ ∧1,1L1).
On the other hand, there are no hypothesis about integrability of V 1,0− , hence
dL1 : Γ(∧1,0L1) −→ Γ(∧0,2L1 ⊕ ∧1,1L1 ⊕ ∧2,0L1).
Using the Leibniz rule, we have
dL1 : Γ(∧k,lL1) −→ Γ(∧k+1,lL1 ⊕ ∧k,l+1L1 ⊕ ∧k−1,l+2L1).
One can easily check that N is just the Nijenhuis tensor corresponding to the complexstructure J 2|L1 , i.e.,
N ∈ ∧2V 0,1− ⊗ V 1,0
+ , N (X,Y ) = −[[X,Y ]]+, for all X,Y ∈ Γ(V 1,0− ),
where ·+ indicates projection onto V+ and this is precisely how the operator N introducedin (3.2) is defined, hence these two operators agree.
Finally, for α ∈ Γ(∧k,0L1) = Γ(∧kV 0,1+ ), πk+1,0 dL1α is determined by its values on
k + 1 elements v0, · · · , vk ∈ V 1,0+ :
(πk+1,0 dL1α)(v0, · · · , vk) = (dL1α)(v0, · · · , vk)
=∑i
(−1)iLπT viα(v0, · · · , vi, · · · , vk)+
+∑i,j
(−1)i+jα([[vi, vj ]], v0, · · · , vi · · · , vj , · · · , vk)
= (∂+α)(v0, · · · , vk),
where we have simply used the definitions of the operators dL1 and ∂+.
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This allows us to define operators
∂+ : Γ(∧k,lL1) −→ Γ(∧k+1,lL1);
∂− : Γ(∧k,lL1) −→ Γ(∧k,l+1L1);
and the condition d2L1
= 0 gives:
Corollary 7.2. Let ∂+, ∂− and N be the operators defined above. Then
∂2+ = N 2 = 0 and ∂
2− = −∂+,N.
Finally, the operator dL1 is related to the operator ∂J 1 via (see [21, 27]),
∂J 1(α · ϕ) = (dL1α) · ϕ+ (−1)|α|α · ∂J 1ϕ.
So, if we take α ∈ Γ(∧k,lL1) and ϕ ∈ Up,q we can split the equation above in threecomponents which give the following relations:
Proposition 7.3. For α ∈ Γ(∧•L1) and ϕ ∈ Ω•(M ;C) we have
δ+, α · ϕ = (∂+α) · ϕ;
δ−, α · ϕ = (∂−α) · ϕ;
N,α · ϕ = N (α) · ϕ.
7.2 Further operators
Proposition 7.3 relates the Lie algebroid operators to the graded commutator of the actionof an element α ∈ Γ(∧•L1) and the operators δ+, δ− and N . In an SKT manifold, onecan define more operators that can be obtained this way but which do not come from Liealgebroids. The reason to consider such operators will be made clear next section whenstudying the deformation and stability problem. There we will see that the correct bundleto consider is L2, the +i-eigenspace of J 2. Since L2 is not a Lie algebroid we are forcedto consider differential operators associated to brackets which do not satisfy Jacobi, andhence do not square to zero, but we don’t let this mishap dampen our sentiment.
As before, we let J 1 be a generalized complex extension of the SKT structure andinspired by Proposition 7.3 we define
∂± : Clif(TCM) −→ Diff(Ω•(M ;C)) ∂±α = δ±, α;∂± : Clif(TCM) −→ Diff(Ω•(M ;C)) ∂±α = δ±, α;
N : Clif(TCM) −→ Clif(TCM) Nα = N,α;(7.2)
where Diff(Ω•(M ;C)) is the vector space of differential operators on forms which is itselfa Z2-graded Lie algebra with graded commutator as the bracket.
We denote Γ(∧k,lL2) = Γ(∧kV 0,1+ ⊗ ∧lV 1,0
− ). Next we summarize the main propertiesof these operators.
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Proposition 7.4. The restriction of N to ∧•L2 vanishes and
∂+ : Γ(∧kV 0,1+ ⊗ ∧lV 1,0
− ⊗ ∧mV 0,1− ) −→ Γ(∧k+1V 0,1
+ ⊗ ∧lV 1,0− ⊗ ∧mV 0,1
− ); (7.3)
∂− : Γ(∧k,lL2) −→ Γ(∧k,l+1L2). (7.4)
Further, the following relations and their complex conjugates hold:
∂2+ = N 2
= 0; ∂+,N = 0;
∂2− = −N , ∂+; N , ∂− = 0; ∂−, ∂+ = 0; ∂+, ∂− = −∂−,N;
∂+, ∂++ ∂−, ∂−+ N ,N = 0.
Proof. The relations between the operators follow from the corresponding relations forδ+, δ− and N and the fact that the vector space of operators on forms are a Z2-gradedLie algebra with Lie bracket given by the graded commutator of operators. If we letα ∈ Clif(TCM), using the graded Jacobi identity we have
dH , dH , α = dH , dH, α − dH , dH , α = −dH , dH , α = 0,
where we have used that dH , dH = 2(dH)2 = 0. Hence, splitting dH into its six com-ponents, taking α ∈ Γ(∧jV 1,0
+ ⊗ ∧kV 0,1+ ⊗ ∧lV 1,0
− ⊗ ∧mV 0,1− ), applying this operator to
elements in Up,q and imposing the vanishing of the different components of the result weget the relations stated in the theorem. For example the vanishing of the component inUp+(j−k)+(l−m)+2,q+(j−k)−(l−m) gives
δ−, δ+, α+ δ+, δ−, α+ δ−, N,α+ N, δ−, α = 0.
That is∂−∂+ + ∂+∂− + ∂−N +N∂− = 0.
Which is the same as∂+, ∂−+ ∂−,N = 0
Since N ∈ Γ(∧3L2) and L2 is isotropic, N,α vanishes for every α ∈ Γ(∧•L2).Next we will prove that (7.3) and (7.4) hold. Firstly, we prove that when restricted to
the domains indicated in (7.3) and (7.4) ∂+ and ∂− actually take values in the Cliffordalgebra of TM . Indeed, for α ∈ Clif(TM), ϕ ∈ Up,q and f ∈ C∞(M) we decompose dfinto its V 1,0
+ , V 0,1+ , V 1,0
− and V 0,1− components, say, ∂+f , ∂+f , ∂−f and ∂−f , respectively
and compute
dH , α(fϕ) = dHαfϕ− (−1)|α|αdH(fϕ)
= df ∧ αϕ+ fdHα · ϕ− (−1)|α|α(df ∧ ϕ+ fdHϕ)
= (∂−f + ∂−f + ∂+f + ∂+f) · αϕ+ fdHαϕ
− (−1)|α|α((∂−f + ∂−f + ∂+f + ∂+f) · ϕ+ fdHϕ).
= fdHαϕ+ (∂−f + ∂−f + ∂+f + ∂+f) · αϕ− (−1)|α|α((∂−f + ∂−f + ∂+f + ∂+f) · ϕ).
(7.5)
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If α ∈ Γ(∧kV 0,1+ ⊗ ∧lV 1,0
− ⊗ ∧mV 0,1− ), then ∂+f graded commutes with α and hence if we
compose dH , α with the projection onto Up−k+l−m−1,q−k−l+m−1 we have
πUp−k+l−m−1,q−k−l+m−1 dH , α : Up,q −→ Up−k+l−m−1,q−k−l+m−1
is C∞-linear, that is, it is a tensor. Further, using the decomposition dH into its sixcomponents, as in Theorem 3.6, we see that the tensor above corresponds to:
∂+α = δ+, α : Up,q −→ Up−k+l−m−1,q−k−l+m−1
Since the action of Clif(TM) establishes an isomorphism between Clif(TM) and the spaceof endomorphisms of ∧•T ∗M , the tensor ∂+α corresponds to an element in Clif(TM):
∂+ : Γ(∧kV 0,1+ ⊗ ∧lV 1,0
− ⊗ ∧mV 0,1− ) −→ Clif(TCM)
Similarly, if we require α to lie in Γ(∧k,lL2) the same argument gives us the result for ∂−.Indeed, in this setting, ∂−f also graded commutes with α in (7.5), hence we get anothertensor, this time by composing dH , α with the projection onto Up−k+l−m+1,q−k−l+m−1,that is, we get
∂− : Γ(∧k,lL) −→ Clif(TCM). (7.6)
Now we deal with ∂+.
Lemma 7.5. Let (M,G, I) be a positive SKT manifold and J be a generalized complexextension of the SKT structure. The vector bundle V 1,0
− is a representation of the Lie
algebroid V 1,0+ if endowed with the connection
∇ : Γ(V 1,0+ ⊗ V 1,0
− ) −→ Γ(V 1,0− ); ∇vw = π
V 1,0−
([[v, w]]),
where v ∈ Γ(V 1,0+ ) and w ∈ Γ(V 1,0
− ). That is, for v, v1, v2 ∈ Γ(V 1,0+ ), w ∈ Γ(V 1,0
− ) andf ∈ C∞(M) we have
∇fvw = f∇vw, ∇vfw = (LπT (v)f)w + f∇vw
∇v1∇v2 −∇v2∇v1 −∇[[v1,v2]] = 0.
Proof. The first two identities follow from basic properties of the Courant bracket:
∇fvw = πV 1,0−
([[fv, w]]) = πV 1,0−
(f [[v, w]]− (LπT (w)f)v) = πV 1,0−
(f [[v, w]]) = f∇vw.
∇vfw = πV 1,0−
([[v, fw]]) = πV 1,0−
(f [[v, w]] + (LπT (v)f)w) = f∇vw + (LπT (v)f)w.
Now we prove flatness. Let us first consider the term ∇v1∇v2w. We decompose [[v2, w]]into two components, according to the decomposition L1 = V 1,0
+ ⊕ V 1,0− and ∇v2w is the
component in V 1,0− :
[[v2, w]] = u1,0+ +∇v2w, u1,0
+ ∈ Γ(V 1,0+ ).
Since V 1,0+ is involutive, we see that
[[v1, u1,0+ ]] ∈ V 1,0
+ .
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Hence the V 1,0− component of [[v1,∇v2w]] is the V 1,0
− component of [[v1, [[v2, w]]]], that is
∇v1∇v2w = πV 1,0−
([[v1, [[v2, w]]]]).
Therefore we have
∇v1∇v2w −∇v2∇v1w −∇[[v1,v2]]w = πV 1,0−
([[v1, [[v2, w]]]]− [[v2, [[v1, w]]]]− [[[[v1, v2]], w]]) = 0,
where in the last equality we used the fact that the Courant bracket satisfies the Jacobiidentity.
Since ∇ is a connection on V 1,0− , we can extend it to the exterior algebra of V 1,0
− :
∇ : Γ(∧kV 1,0− ) −→ Γ(V 0,1
+ ⊗ ∧kV 1,0− );
∇vα = v,∇α for all v ∈ Γ(V 1,0+ ).
Lemma 7.6. For α ∈ Γ(∧kV 1,0− ) we have
∂+α = ∇α.
Proof. Indeed, we check that when paired (i.e., graded commuted) with an element v ∈Γ(V 1,0
+ ) these two operators give the same result. Unwinding the definition of ∂+, wehave that for ϕ ∈ Up,q v, ∂+αϕ is the Up+k,q−k component of v, dH , αϕ. For αdecomposable, α = α1 ∧ · · · ∧ αk we have
v, ∂+αϕ = πUp+k,q−k(v, dH , α1 ∧ · · · ∧ αkϕ)
= πUp+k,q−k(k∑i=1
α1 ∧ · · · ∧ [[v, αi]] ∧ · · · ∧ αkϕ)
=
k∑i=1
α1 ∧ · · · ∧ πV 1,0−
[[v, αi]] ∧ · · · ∧ αkϕ
=k∑i=1
α1 ∧ · · · ∧ v,∇αi ∧ · · · ∧ αkϕ
= v,∇αϕ,
where in the second equality we used that the Courant bracket is the derived bracket asso-ciated to dH hence the commutator becomes a Schouten–Nijenhuis type bracket betweenthe sections involved.
Since ∂+α and ∇α are both elements in Clif(TCM) we have just established that theirdifference lies in the annihilator of V 1,0
+ under graded commutator, that is
∇α− ∂+α ∈ Γ(∧•V 1,0+ ⊗ ∧•V−).
And we have that when acting in Up,q
∇α− ∂+α : Up,q −→ Up+k−1,q−k−1 ⊂ Wp+q−2
as individually both ∂+α and ∇α have this property. On the other hand, elements in∧lV 1,0
+ ⊗ ∧mV− map Up,q ⊂ Wp+q into Wp+q+l, so the difference ∇α − ∂+α vanishes andthe result follows.
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Proof of (7.3). Now, let αk,l ∈ Γ(∧kV 0,1+ ⊗ ∧lV 0,1
− ) = Γ(∧k,lL1) and αm,0− ∈ Γ(∧mV 1,0− ).
Then for a form ϕ we have
∂+(αk,lαm,0− ) · ϕ = δ+, αk,lαm,0− ϕ
= δ+(αk,lαm,0− ϕ) + (−1)k+l+mαk,lαm,0− δ+ϕ
= (∂+αk,l)αm,0− ϕ+ (−1)k+lαk,lδ+(αm,0− ϕ) + (−1)k+l+mαk,lαm,0− δ+ϕ
= (∂+αk,l)αm,0− ϕ+ (−1)k+lαk,l(∂+α
m,0− )ϕ
= ((∂+αk,l)αm,0− + (−1)k+lαk,l(∂+α
m,0− ))ϕ.
According to Proposition 7.3, (∂+αk,l)αm,0− ∈ Γ(∧k+1V 0,1
+ ⊗∧lV0,1− ⊗∧mV
1,0− ) and, according
to Lemma 7.6, αk,l(∂+αm,0− ) also belongs to this space, hence we have proved (7.3).
Proof of (7.4). Since L2 is maximal isotropic, ∧•L2 ⊂ Clif(TCM) can be characterized as
∧•L2 = β ∈ Clif(TCM) : β, v = 0 for all v ∈ L2.
Now, given α ∈ Γ(∧k,lL2), we already know that
∂−α : Up,q −→ Up−k+l+1,q−k−l−1 (7.7)
so to prove that ∂−α ∈ Γ(∧k,l+1L2) it is enough to prove that ∂−α ∈ Γ(∧•L2) as the onlyelements in Γ(∧•L2) with property (7.7) are those in Γ(∧k,l+1L2). So, for α ∈ Γ(∧k,lL2),v ∈ Γ(L2) and ϕ ∈ Up,q we compute
∂−α, vϕ = δ−, α, vϕ= πUp−k+l,q−k−l−2(dH , α, vϕ)
= ±πUp−k+l,q−k−l−2([[α, v]]SNϕ)
= 0,
where in the third equality we used that L2 is isotropic, and hence the derived bracketis just the extension of the Courant bracket to a Schouten–Nijenhuis-like bracket and inthe last equality we have used that we have used that the Schouten–Nijenhuis bracket hasdegree −1, hence it maps (v, α) ∈ L2 × ∧k,lL2 into Clifk+l(TCM) and hence it can notmap Up,q to Up−k+l,q−k−l−2 as elements in Clifj(TCM) can only change the first and thesecond index of the bigrading of forms (the ‘p’ and the ‘q’) by at most ±j.
As a consequence of Proposition 7.4, we see that although L2 is not a Lie algebroid, wecan still define some cohomology there using ∂+.
Definition 7.7. Let (G, I) be a positive SKT structure on a manifold with 3-form (M,H)and J be a generalized complex extension of I. The ∂+-cohomology in L2, H•,•
∂+(M ;L2),
is the quotient
Hk,l
∂+(M ;L2) =
ker(∂+ : Γ(∧k,lL2) −→ Γ(∧k+1,lL2))
Im (∂+ : Γ(∧k−1,lL2) −→ Γ(∧k,lL2))
48
Finally we observe that all the operators we have studied in this section where obtainedwith the help of a generalized complex extension of the SKT structure, so, they are notintrinsic to the SKT structure. Yet, one could follow the same road and define
∂N+ : Clif(TCM) −→ Diff(Ω•(M ;C)) ∂N+α = δN+ , α;
d− : Clif(TCM) −→ Diff(Ω•(M ;C)); d−α = /D−, α.
By choosing a generalized complex extension to I, the decomposition δN+ = δ+ +N and/D− = δ− + δ− we see that
∂N+ = ∂+ +N d− = ∂− + ∂−.
And as a consequence of Corollary 3.4 and Propositions 7.3 and 7.4 we have
Proposition 7.8. Let (G, I) be a positive SKT structure on a manifold with closed 3-form
(M,H). Then the operators ∂N+ and d− satisfy
(∂N+ )2 = 0; ∂N+ , d− = 0; ∂N+ , ∂N+ + d2− = 0;
and
∂N+ : Γ(∧kV 0,1+ ⊗ Clif(V−)) −→ Γ(∧k+1V 0,1
+ ⊗ Clif(V−));
d− : Γ(∧kV 0,1+ ) −→ Γ(∧kV 0,1
+ ⊗ V−).
7.3 Representations
We finish our study of Lie algebroids and related objects associated to an SKT structurethis section with some remarks about representations of the Lie algebroid V 1,0
+ .
Firstly, it is worth pointing out that projection πT : V 1,0+ −→ TM has as its image
T 1,0M , the +i-eigenspace of the complex structure introduced in Proposition 3.5 and henceV 1,0
+ is isomorphic to T 1,0M as a Lie algebroid and a representation of one is the same asa representation of the other. Of course representations of T 1,0M are simply holomorphicbundles endowed with a holomorphic connection (in fact, the connection determines theholomorphic structure), so one may just as well replace the words “representation of V 1,0
+ ”by “holomorphic vector bundle”.
In Lemma 7.5 we saw that V 1,0− is a representation of V 1,0
+ . In the proof we have used
the fact that J 1 is integrable, but that is not entirely necessary and in fact V 0,1− is also a
representation of V 1,0+ .
Proposition 7.9. Let (M,G, I) be a positive SKT manifold and J be a generalized complexextension of the SKT structure. The vector bundle V 0,1
− is a representation of the Lie
algebroid V 1,0+ if endowed with the connection
∇′ : Γ(V 1,0+ ⊗ V 0,1
− ) −→ Γ(V 0,1− ); ∇′vw = π
V 0,1−
([[v, w]]), ∀v ∈ Γ(V 1,0+ ), w ∈ Γ(V 0,1
− ).
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Proof. The proof that ∇′ is a connection is the same as that given for ∇ in Lemma 7.5.So we only prove flatness, i.e.,
∇′v1∇′v2 −∇
′v2∇
′v1 −∇
′[[v1,v2]] = 0 for all v1, v2 ∈ Γ(V 1,0
+ ).
Again, we first consider the term ∇′v1∇′v2w. We decompose [[v2, w]] into four components,
according to the decomposition TCM = V 1,0+ ⊕ V 0,1
+ ⊕ V 1,0− ⊕ V 0,1
− . Since the failure ofL2 to be involutive with respect to the Courant bracket is given by the Nijenhuis tensor,N ∈ V 0,1
+ ⊗∧2V 1,0− , we see that the component of [[v2, w]] outside L2 is given by −N(v2, w) ∈
V 1,0− , so [[v2, w]] has only three components and ∇′v2w is the component in V 0,1
− :
[[v2, w]] = u1,0+ +∇′v2w −N(v2, w), u1,0
+ ∈ Γ(V 1,0+ ).
Since V 1,0+ ⊂ L1 and L1 is integrable, we have that
[[v1, u1,0+ −N(v2, w)]] ∈ L1
hence the V 0,1− component of [[v1,∇′v2w]] is the V 0,1
− component of [[v1, [[v2, w]]]], that is
∇′v1∇′v2w = π
V 0,1−
([[v1, [[v2, w]]]]).
Therefore we have
∇′v1∇′v2w −∇
′v2∇
′v1w −∇
′[[v1,v2]]w = π
V 0,1−
([[v1, [[v2, w]]]]− [[v2, [[v1, w]]]]− [[[[v1, v2]], w]]) = 0,
where the last equality follows from the Jacobi identity.
Finally, V−⊗C ∼= TCM itself is a representation of V 1,0+ . Of course, V−⊗C = V 1,0
− ⊕V0,1−
and since both summands are representations of V 1,0+ , so is the sum. Yet, the natural
connection one should consider on V−⊗C is not the sum of the connections, but it shouldalso include the Nijenhuis tensor of J 2. This is also Bismut’s “exotic holomorphic structureon TCM” [2].
Proposition 7.10. Let (M,G, I) be a positive SKT manifold. The vector bundle V−C =V− ⊗ C is a representation of the Lie algebroid V 1,0
+ if endowed with the connection
∇− : Γ(V 1,0+ ⊗V−C) −→ Γ(V−C); ∇vw = πV−C([[v, w]]), for all v ∈ Γ(V 1,0
+ ), w ∈ Γ(V−C).
Further, if J is a generalized complex extension of the SKT structure, we can split V−C =V 1,0− ⊕ V 0,1
− and in this splitting ∇− is given by
∇− =
(∇ −N0 ∇′
)(7.8)
where ∇ is the connection introduced in Lemma 7.5 and ∇′ is the connection introduced inProposition 7.9
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Proof. First we prove that in the presence of a generalized complex extension the definitionof ∇− agrees with the expression given in (7.8). Given v ∈ Γ(V 1,0
+ ), w1,0 ∈ Γ(V 1,0− )
and w0,1 ∈ Γ(V 0,1− ) we have that [[v, w1,0]] lies in L1 by integrability of J 1 hence its V−
component is its V 1,0− component which is, by definition, ∇vw1,0.
Similarly, [[v, w0,1]] has three components, as established in the previous Proposition:
[[v, w0,1]] = u+ +∇′vw0,1 −N(v, w0,1),
hence, projecting onto V− we get
∇−v w0,1 = ∇′vw0,1 −N(v, w0,1)
and hence ∇− is given by (7.8)Further, similarly to the previous case, we have that for v1, v2 ∈ Γ(V 1,0
+ )
[[v2, w1,0+w0,1]] = u1,0
+ +∇vw1,0+∇′vw0,1−N(v, w0,1) = u1,0+ +∇−v (w1,0+w0,1), u1,0
+ ∈ Γ(V 1,0+ ).
And again, since V 1,0+ is involutive, we see that
∇−v1∇−v2w = πV−([[v1, πV−([[v2, w]])]]) = πV−([[v1, [[v2, w]]]]),
and flatness of the connection follows once again from the Jacobi identity for the Courantbracket.
With these representations at hand, one can also see the operator ∂+ as studied inProposition 7.4 as an extension of the Lie algebroid differential ∂+ : Γ(∧kV 0,1
+ ) −→Γ(∧k+1V 0,1
+ ) to take values on the tensor product of the exterior algebra of the flat modules
V 1,0− and V 0,1
− . Similarly, the operator ∂N+ is the extension of the Lie algebroid differential∂+ to take values in the representation from Proposition 7.10.
8 Deformations
One can consider at least three different deformation problems. In the most strict of themone fixes the generalized metric and asks for deformations of the complex structure onV+. For the second deformation problem, one fixes the Courant algebroid where the SKTstructure is defined and looks for deformations of metric and complex structure which arestill SKT. In the third, and more general problem, one lets the Courant algebroid changeas well, that is, if ωt is the 1-parameter family of Hermitian forms, one only requires thatddctωt = 0, but the cohomology class [dctωt] is allowed to vary with t.
Here we will consider the first two problems. In both cases, we want to find involutivedeformations, V 1,0ε
+ , of V 1,0+ , in the first problem we require V 1,0ε
+ to lie in V+, while in thesecond problem that condition is removed. Either way, as long as the deformation is smallenough, the remaining SKT conditions, namely,
V 1,0ε+ ∩ V 1,0ε
+ = 0
and〈v, v〉 > 0 for all v ∈ V 1,0ε
+ ⊕ V 1,0ε+ , v = v
will still hold, since these are open conditions and hence if V 1,0ε+ is involutive, it defines an
SKT structure.
51
8.1 First deformation problem
The first problem is very similar to the problem of deformations of generalized complexstructures. Indeed, since V+ is to remain fixed, any small deformation of V 1,0
+ is still
transversal to V 0,1+ and hence can be described by the graph of a bundle map ε : V 1,0
+ −→V 0,1
+ :
V 1,0ε+ := v1,0 + ε(v1,0) : v1,0 ∈ V 1,0
+ .
Since V 0,1+∼= (V 1,0
+ )∗, ε is a section of ⊗2V 0,1− . Since V 1,0ε
+ is isotropic, we have that
ε ∈ Γ(∧2V 0,1+ )1.
Theorem 8.1. Let (M,H) be a manifold with 3-form and (G, I) be a positive SKT struc-ture on (M,H). An element ε ∈ Γ(∧2V 0,1
+ ) of small norm gives rise to an integrable SKTstructure Iε compatible with the metric G if and only if
d−ε = 0;
∂N+ ε+1
2[[ε, ε]]SN = 0,
(8.1)
where [[·, ·]]SN is the Schouten–Nijenhuis bracket extending the Courant bracket to Γ(∧•V 0,1+ ).
If J is a generalized complex extension of the SKT structure, the equations above become
∂−ε = ∂−ε = 0
∂+ε+1
2[[ε, ε]]SN = 0.
Proof. For the deformation given by ε ∈ Γ(∧2V 0,1+ ) ⊂ Clif(TCM) as above, the space of
forms which annihilate V 1,0ε+ ,Wn
ε , is given byWnε = e−ε ·Wn and integrability is equivalent
to
dH :Wnε −→Wn
ε ⊕Wn−2ε .
But, since V+ and V− do not change under the deformation, elements in Wnε ⊕Wn−2
ε arethose which can be obtained from Wn
ε by the Clifford action of elements in V+ ⊗Clif(V−)and Wn
ε = e−ε · Wn , so the condition above holds if and only if for all ϕ ∈ Wn there isα ∈ V+ ⊗ Clif(V−) such that
dH(e−ε · ϕ) = α · e−ε · ϕ = e−ε · ((eε∗α) · ϕ).
And since ε ∈ ∧2V 0,1+ eε∗α ∈ V+ ⊗ Clif(V−), as eε only acts in the V+ part as a complex
orthogonal transformation. Therefore, integrability of the deformed structure is equivalentto
eεdHe−ε(ϕ) = α · ϕ for all ϕ ∈ Wn, for some α ∈ Γ(V+ ⊗ Clif(V−)).
Now we can expand the operator on the left as a series:
eεdHe−ε(ϕ) = dH + [ε, dH ] +1
2![ε, [ε, dH ]] +
1
3![ε, [ε, [ε, dH ]]] + · · ·
52
The third term of the series is computed in terms of the derived bracket of ε, with itself,that is, the extension of the Schouten–Nijenhuis bracket to sections of the Lie algebroidV 0,1
+ :
[ε, [ε, dH ]] = ε, ε, dH = −ε, dH, ε = −[[ε, ε]]SN ∈ Γ(∧3V 0,1+ ).
Since V 0,1+ is isotropic, all terms of the series after the third vanish and we have
eεdHe−ε(ϕ) = (dH + [ε, dH ]− 1
2[[ε, ε]]SN )ϕ.
Since the original SKT structure is integrable, we have that dH : Wn −→ Wn ⊕ Wn−2.
Splitting dH = /DH− + δN+ + δN+ we have that [ε, δN+ ] maps Wn into Wn−2. So integrability
of the deformed structure is equivalent to the requirement that the remaining terms alsomap Wn into Wn ⊕Wn−2.
Now notice that [ε, /DH− ] :Wn −→Wn−4 while [ε, δN+ ]− 1
2 [[ε, ε]]SN :Wn −→Wn−6, so forthe deformed structure to be integrable, each of these maps must vanish. From Proposition7.8, these conditions are equivalent to the conditions presented in the statement of thetheorem.
8.2 Second deformation problem and stability
As we mentioned earlier, in the second deformation problem, one fixes the Courant alge-broid and looks for deformations of the SKT structure. Stated this way, the problem hasan infinite dimensional space of solutions even after taking the quotient by the action ofthe diffeomorphism group. Indeed, from the classical viewpoint, an SKT structure is aHermitian structure (g, I) for which ddcω = 0. For any f ∈ C∞(M) we can consider anew Hermitian form: ω + εddcf . In a compact manifold if we take ε small enough thisgives rise to new non diffeomorphic SKT structures on M still with 3-form dcω. In fact,more can be said about the set of deformations of an SKT structure which preserve thecomplex structure I. The author has learnt from Swann and Zabzine a small variation ofthe following proposition, which can also be found, rephrased, in [11]:
Proposition 8.2. The set of small deformations of an SKT structure (g, I) which preservethe complex structure and the cohomology class [dcω] is parametrized by the kernel of themap
dc : Ω1,1(M) −→ Ω3(M)/Ω3exact(M).
In particular there is a disc around the zero form in the set of closed 2-forms on M withnontrivial (1, 1) component, D ⊂ Ω2
cl(M)/(Ω2,0cl (M)+Ω0,2
cl (M)) which maps injectively intothe kernel of the map above.
Proof. The first claim is clear: a variation of the metric is equivalent to a deformationof the Hermitian form ω = g I and since I is orthogonal with respect to the deformedstructure, ω must change by a form B ∈ Ω1,1(M). Then the SKT condition gives
dc(ω +B) = H + dcB,
and since this is to have the same cohomology class as H, dcB must be exact.
53
Now, given any closed 2-form B, let Bp,q be its component in Ωp,q(M). Then letω = ω + B1,1. As long as B1,1 is small, nondegeneracy and positivity of g = −ω I stillhold as those are open conditions. And finally,
dc(ω +B1,1) = H − i∂B1,1 + i∂B1,1 = H + i∂B2,0 − i∂B0,2
= H + d(iB2,0 − iB0,2) = H + 2dIm(B2,0).
The fact that one can so freely deform the metric and remain in the SKT category is avery useful phenomenon which, for example, allows one to blow-up complex submanifoldsof SKT manifolds [13]. Yet, from the point of view of deformations, these seem a little lessinteresting than deformations which change the complex structure.
Therefore it is natural to phrase the second deformation problem as a stability problem.In our setup, the direct translation of the classical question is as follows: Given an SKTstructure (G, I), Proposition 3.5 gives us an extension to a generalized Hermitian structure(G,J 1) for which J 1 is of complex type. Then one is interested in which deformationsof J 1 as a generalized complex structure of complex type give rise to deformations of theSKT structure.
What we do next is phrased in slightly more general terms in order to accommodateother cases of interest. Our proof is inspired on the approach used by Goto to prove thegeneralized Kahler stability theorem [16].
Theorem 8.3. (SKT Stability Theorem) Let (G,J 1) be a generalized Hermitian exten-sion of a positive SKT structure. Given J 1ε, an analytic family of deformations of J 1
parametrized by a small disc D ⊂ C, ε : D −→ Γ(∧2L1), we let
ε+ : D −→ Γ(∧2V 0,1+ )
ε± : D −→ Γ(V 0,1+ ⊗ V 0,1
− )
ε− : D −→ Γ(∧2V 0,1− )
be the components of ε and for each of them let ε• =∑∞
k=11k!ε•ktk be the corresponding
series expansion. Then the obstructions to finding a generalized metric Gε which makes(Gε,J 1ε) into a generalized Hermitian extension of an SKT structure lie in the spaceH2,1
∂+(M ;L2) and the first obstruction is the class [∂−ε
+1 +N ε±1 ].
According to the deformation theorem for Lie bialgebroids [21, 27], a deformation ofJ 1 is given by sections ε(t) as above, which we will denote simply by ε, satisfying theMaurer–Cartan equation and the corresponding deformed structure is given by
J 1ε = IεJ 1I−1ε ,
where Iε is the real transformation which in the splitting TCM = L1 ⊕ L1 is given by thematrix
Iε =
(Id εε Id
).
54
Since this matrix is not orthogonal with respect to the natural pairing, in general, neitheris IεJ 2I
−1ε and hence a complex structure on TM defined this way is not even an almost
generalized complex structure. The first step is to translate this problem into deforma-tions using orthogonal transformations so that we can find an generalized almost complexstructure J 2ε making J 1ε into a generalized Hermitian structure.
Lemma 8.4. There is a map a : D ⊂ ∧2L1 −→ ∧2(L1 ⊕ ∧2L1)R, where D is a small discaround the origin and (·)R denotes the real elements in the vector space, such that for eachε ∈ Γ(∧2L1), a(ε) is the unique element in the codomain that satisfies
ea∗J 1e−a∗ = IεJ 1I
−1ε .
Further a∗|0(ε) = −(ε+ ε).
Proof. Indeed, the space of generalized complex structures of the same parity as J 1 on TpMis the homogeneous space SO(TpM)/Stab(J 1) ∼= SO(n, n)/U(n, n). Hence, composing theexponential map with the projection
so(TpM) −→ SO(TpM) −→ SO(TpM)/Stab(J 1)
gives a local submersion. Since the elements in so(TpM) preserving J 1 are those in (L1⊗L1)R, and (∧2L1 ⊕ ∧2L1)R is a complementary subspace, we have that
(∧2L1 ⊕ ∧2L1)R −→ SO(TpM)/Stab(J 1).
is a local diffeomorphism. That is, for each small deformation of a generalized complexstructure J 1 on V there is a unique element a ∈ (∧2L1 ⊕ ∧2L1)R which realizes it.
For the last claim, we observe that if ε : (−s, s) −→ ∧2L1 is a path which passesthrough 0 at time zero and v ∈ L1, then Iε|0(v + v) = ε|0(v) + ε|0(v). On the other hand,for a : (−s, s) −→∈ (∧2L1 ⊕ ∧2L1)R, say a = α+ α with α ∈ ∧2L1, we have
ddte
a∗|0(v + v) = d
dt(ea(v + v)e−a)|0 = a|0, v + v = −(α|0(v) + α|0(v)),
hence if the family a gives rise to the same deformation as ε, we must have a = −(ε+ε).
Proof of Theorem 8.3. With this setup, ea∗ is an orthogonal transformation of TM andhence ea∗J 2e
−a∗ is an generalized almost complex structure which commutes with J 1ε. If
b ∈ R⊕ (L1⊗L1)R ⊂ Clif2(TM) the orthogonal transformation eb∗ preserves J 1 hence J 1ε
and J 2ε = ea∗eb∗J 2e
−b∗ e−a∗ give rise to a generalized Hermitian structure for any choice of
b. Our task is to identify suitable conditions on a given a under which we can choose b sothat the pair J 1ε and J 2ε gives an SKT structure.
Since J 1ε is integrable, we know that
dH : U0,nε −→ U1,n−1
ε ⊕ U1,n−3ε ⊕ U−1,n−1
ε ⊕ U−1,n−3ε
and, according to Theorem 3.6, the SKT condition is that the U−1,n−3ε component of the
map above must vanish.
55
Since J 1ε and J 2ε are obtained from J 1 and J 2 by the action of eaeb ∈ Spin(TM),the corresponding decompositions of spinors are related by
Up,qε = eaebUp,q,
so the different components of dH in the splitting induced by (J 1ε,J 2ε) (c.f. Figure 3)
dH = δ+ε + δ+ε + δ−ε + δ−ε +Nε +Nε +N2ε +N2ε
are in correspondence with the components of e−be−adHeaeb in the splitting determinedby (J 1,J 2) (see Figure 12).
U0,n
e−ae−bδ+εeaeb
ww
e−ae−bN2εeaeb
e−ae−bδ−εeaeb
&&
e−ae−bNεeaeb
U−1,n−1 U1,n−1
U0,n−2
. . . U−1,n−3 U1,n−3 . . .
Figure 12: Nontrivial components of e−be−adHeaeb and their relation to the components of dH
with respect to the decomposition of forms determined by (J 1ε,J 2ε).
Since a ∈ (∧2L1 ⊕ ∧2L1)R, there are α+ ∈ Γ(∧2V 0,1+ ), α± ∈ Γ(V 0,1
+ ⊗ V 0,1− ) and α− ∈
Γ(∧2V 0,1− ) such that
a = α+ + α± + α− + α+ + α± + α−. (8.2)
Similarly, any b ∈ Γ(R ⊕ L1 ⊗ L1)R can be decomposed in four components according tothe splitting
R⊕ (L1 ⊗ L1)R = R⊕ (V 1,0+ ⊗ V 0,1
+ )R ⊕ (V 1,0− ⊗ V 0,1
− )R ⊕ ((V 0,1+ ⊗ V 1,0
− )⊕ (V 1,0+ ⊗ V 0,1
− )R)
however, as we will see later, only the last of these components is relevant for the defor-mation problem. So we let b is of the form
b = β + β± + β± (8.3)
with β± ∈ Γ(V 0,1+ ⊗ V 1,0
− ) and β ∈ Γ(R⊕ (V 1,0+ ⊗ V 0,1
+ )R ⊕ (V 1,0− ⊗ V 0,1
− )R).
Since the deformation family ε(t) is analytic on t, so is a(t) and we can try to solvethe condition that the U−1,n−3 component of e−be−adHeaebψ vanishes by a power seriesargument. In the sequence given an analytic function f(t), where f can be α+, α± andso on, we denote denote the coefficients of its power series expansion by fk, so that f =∑ 1
k!fktk.
56
The first nontrivial condition on a imposed by the integrability of J 1 is that [dH , a1]must have no components in Un−3,•. In terms of the decomposition of a above, this implies
[δ+, α+1 ] = 0 [δ+, α
±1 ] + [δ−, α
+1 ] = 0
or equivalently∂+α
+1 = 0 ∂+α
±1 + ∂−α
+1 = 0. (8.4)
The first term in the series expansion of e−be−adHeaebψ is
[dH , b1 + a1]ψ.
Splitting a and b into their components, we see that the requirement that the U−1,n−3
component vanishes is equivalent to
([δ+, β±1 ] + [δ−, α
+1 ] + [N,α±1 ])ψ = 0
Isolating β± and using that ψ is nonzero, we can rewrite the condition above as
∂+β±1 = −∂−α+
1 −Nα±1 .
And one obvious necessary condition for this to have a solution is that the right hand sidemust be ∂+-closed. That is indeed the case, since
∂+(∂−α+1 +Nα±1 ) = −∂−∂+α
+1 − ∂−Nα
+1 −N∂−α
+1 −N∂+α
±1 = 0
where for the first equality we used the commutation rule for ∂+ and ∂− from Proposition7.4 and in the second equality we used equations (8.4) and the fact that Nα+
1 = 0 asα+
1 ∈ L2. If this ∂+-closed form is in fact ∂+-exact, we can use G∂+ , the Green operator
for ∂+, to find a suitable β±1 :
β±1 = −∂∗+G∂+(∂−α+1 +Nα±1 )
Finally, we recall that according to Lemma 8.4, a1 = −(ε1 + ε1), so the components ofa1 are determined by the different components of ε1 hence the equation β±1 must solve is
∂+β±1 = ∂−ε
+1 +N ε±1 ,
so the first obstruction class is [∂−ε+1 +N ε±1 ] ∈ H2,1
∂+(M ;L2).
Now we move to the general case, which is proved by induction. We assume that wehave chosen ai and bi for i < k such that the component of e−be−adHeaeb mapping U0,n
into U−1,n−3 vanishes to order k− 1. Then the vanishing of the order k component of thismap is the condition
[β±k , δ+] + [α+k , δ−] + [α±k , N ] + Fk(a1, · · · , ak−1, b1, · · · , bk−1) = 0,
where Fk is some function. This condition is equivalent to
∂+β±k = −(∂−α
+k +Nα±k + Fk(a1, · · · , ak−1, b1, · · · , bk−1)).
57
And if there is β±k which solves this equation, we can proceed to the next step. If for allpossible choices of bi for i < k which guarantee the vanishing of the lower order terms thisequation has no solution, then the deformation is obstructed.
Next we show that ∂−α+k + Nα±k + Fk(a1, · · · , ak−1, b1, · · · , bk−1) is ∂+-closed so we
can conclude that the obstruction space is H2,1(M ;L2). Since ∂+β±k is clearly ∂+-closed,
we must prove that
∂+(∂+β±k + ∂−α
+k +Nα±k + Fk(a1, · · · , ak−1, b1, · · · , bk−1)) = 0
but the term in parenthesis is precisely the order k component of e−be−aN2εeaeb (see Figure
12) and by assumption, the lower order terms of e−be−aN2εeaeb vanish, therefore we must
prove that
∂+e−be−aN2εe
aeb = δ+, e−be−aN2εe
aeb (8.5)
vanishes to order k. But since δ+ = e−be−aδ+εebea + O(t), we see that (8.5) vanishes to
order k if and only if
e−be−aδ+εeaeb, e−be−aN2εe
aeb
vanishes to order k. But for this operator, we have
e−be−aδ+εeaeb, e−be−aN2εe
aeb = e−be−aδ+ε, N2εeaeb = 0,
where in the last equality we used that, in a generalized Hermitian manifold, the condition(dH)2 = 0 gives, among other things, that δ+, N2 = 0.
Finally, if all obstructions vanish, standard elliptic estimates show that the sequenceconstructed above converges. See for example [16] for the proof of convergence for theanalogous problem in the generalized Kahler case.
It is interesting to notice that one can deform the generalized complex extension J 1
without deforming the SKT structure.
Proposition 8.5. Let (G,J 1) be a generalized complex extension of a positive SKT struc-ture (G, I) on a manifold M . Let ε ∈ Γ(∧2V 0,1
− ) ⊂ Γ(∧2L1) be a Maurer–Cartan element,i.e., the corresponding deformation of J 1, J 1ε, is integrable. Then, if ε is small enough,(G,J 1ε) is a generalized complex extension of the SKT structure (G, I).
Proof. According to Lemma 8.4, if ε is small enough, there is an element a ∈ Γ(∧2L1 ⊕∧2L1)R such that
e−a∗ J 1ea∗ = e−ε∗ J 1e
ε
but since ε ∈ ∧2V 0,1− one can easily see that a ∈ Γ(∧2V 1,0
− ⊕ ∧2V 0,1− ). Then ea|V+ = Id
and ea|V− is an orthogonal transformation of V−, therefore e−aGea = G and e−aJ 1ea|V+ =
J 1|V+ = I. That is e−aJ 1ea is a generalized complex extension of (G, I).
Of course, the same arguments with the obvious changes give deformation results fornegative SKT structures (c.f. Figure 8):
58
Theorem 8.6. (Negative SKT Stability Theorem) Let (G,J 1) be a generalized Hermitianextension of a negative SKT structure. Given J 1ε, an analytic family of deformations ofJ 1 parametrized by a small disc D ⊂ C, ε : D −→ Γ(∧2L1), we let
ε+ : D −→ Γ(∧2V 0,1+ )
ε± : D −→ Γ(V 0,1+ ⊗ V 0,1
− )
ε− : D −→ Γ(∧2V 0,1− )
be the components of ε and for each of them let ε• =∑∞
k=1 ε•ktk be the corresponding
series expansion. Then the obstructions to finding a generalized metric Gε which makes(Gε,J 1ε) into a generalized Hermitian extension of a negative SKT structure lie in the
space H2,1∂−
(M ;L2) and the first obstruction is the class [∂+ε−1 +N 2ε
±1 ].
Proposition 8.7. Let (G,J 1) be a generalized complex extension of a negative SKT struc-ture (G, I) on a manifold M . Let ε ∈ Γ(∧2V 0,1
+ ) be a Maurer–Cartan element and J !ε bethe corresponding deformation of J 1. Then, if ε is small enough, (G,J 1ε) is a generalizedcomplex extension of the SKT structure (G, I).
8.3 Deformations of generalized Kahler structures
Similarly to the SKT deformation problem studied in Section 8.2, a common way to studythe question of deformations of a (generalized) Kahler structure (J 1,J 2) is in the contextof stability: For which deformations of J 1 is there a corresponding deformation of J 2 suchthat the pair of deformed structures is still a (generalized) Kahler structure?
In its classical setting, this question was successfully answered by Kodaira [24], whereJ 1 is taken to be the complex structure and in the generalized setting, for analytic defor-mations, by Goto [16] under the additional hypothesis that the canonical bundle of J 2 hasa globally defined nowhere vanishing closed section. In both cases, there are no obstruc-tions to the problem. However, even in the Kahler setting, if we let J 1 be the symplecticstructure and J 2 be the complex, the problem does have obstructions [8].
As we have seen in Section 2, a generalized Kahler structure is at the same time apositive and a negative SKT structure. Hence, when studying the stability problem of ageneralized Kahler structure there are three possible outcomes one can expect:
1. The deformation is unobstructed and one can complete the deformed J 1 to a gener-alized Kahler structure;
2. The deformation is half obstructed and one can complete the deformed J 1 to apositive or negative SKT structure;
3. The deformation is fully obstructed and J 1 can not be completed to either a positiveor a negative SKT structure.
Theorem 8.8 (Generalized Kahler Stability Theorem). Let (M,J 1,J 2) be a compact gen-eralized Kahler manifold. Given J 1ε, an analytic family of deformations of J 1 parametrized
59
by a small disc D ⊂ C, ε : D −→ Γ(∧2L1), we let
ε+ : D −→ Γ(∧2V 0,1+ )
ε± : D −→ Γ(V 0,1+ ⊗ V 0,1
− )
ε− : D −→ Γ(∧2V 0,1− )
be the components of ε and for each of them let ε• =∑∞
k=11k!ε•ktk be the corresponding
series expansion. Then the obstructions to finding a generalized complex structure J 2ε
which makes (J 1ε,J 2ε) into a generalized Kahler structure lie in the space
H(2,1)+(1,2) =ker(dL2 : Γ(∧2,1L2 ⊕ ∧1,2L2))
Im (dL2 : Γ(∧1,1L2)),
and the first obstruction class is [∂+ε−1 + ∂−ε
+1 ].
Proof. According to Lemma 8.4, there is an element a ∈ Γ(∧2L1 ⊕ ∧2L1)R such that
e−a∗ J 1ea∗ = e−ε∗ J 1e
−ε
and, as in Theorem 8.3, one can compose the orthogonal transformation ea∗ with an or-thogonal tranformation eb∗ which preserves J 1 and still obtain the same deformation, thatis, for any b ∈ Γ(L1 ⊗ L1)R, J 1ε and J 2ε are given by
J 1ε = e−a∗ J 1ea∗ and J 2ε = e−ae−bJ 2e
bea.
As before, our quest is to find b for which J 2ε is integrable and once again we do so using apower series argument. The canonical bundle of J 2ε is given by eaebU0,n and integrabilityis equivalent to
e−be−adHeaeb : U0,n −→ Un−1,1 ⊕ Un−1,−1.
If we decompose a and b into their V 1,0± and V 0,1
± components as in (8.2) and (8.3) andwrite the corresponding components in power series, then the first condition we must solveis the linear vanishing of the components of NJ 2ε, the Nijenhuis tensor of J 2ε
e−be−adHeaeb : U0,n −→ U1,n−3 ⊕ U−1,n−3.
The linear term which maps U0,n into U−1,n−3 ⊕ U1,n−3 is
dL2β±1 + ∂−α
+1 + ∂+α
−1
So we see that the vanishing of the linear part of the Nijenhuis tensor is equivalent thefollowing condition
dL2β±1 = −(∂+α
−1 + ∂−α
+1 ). (8.6)
Integrability of J 1ε means that the U±3,n−3 components of e−ae−bdHeaeb|U0,n vanish and
the linear part of these components is ∂+α+1 : U0,n −→ U−3,n−3 and ∂−α
−1 : U0,n −→
60
U3,n−3, hence α+1 is ∂+-closed and α−1 is ∂−-closed, showing that ∂+α
−1 + ∂−α
+1 is dL2-
closed and hence represents a class in H(2,1)+(1,2). If this class vanishes, we can find β±1which solves (8.6). That is the first obstruction class is
[∂+ε−1 + ∂−ε
+1 ] ∈ H(2,1)+(1,2).
Next we assume that we have chosen β±j for j < k such that the Nijenhuis tensor of
J 2 vanishes to order k − 1 and we must choose β±k which makes the Nijenhuis tensor ofJ 2 vanish to order k. Expanding the operator e−be−adHeaeb to order k we see that thedegree k part of the Nijenhuis tensor is:
(NJ 2ε)k = dL2β±k + ∂−α
+k + ∂+α
−k + Fk(a1, · · · , ak−1, b1, · · · , bk−1), (8.7)
for some function Fk which takes values in Γ(∧2,1L2 ⊕∧1,2L2), and the vanishing of NJ 2ε
to order k is equivalent to the requirement that the term above vanishes, which can onlybe achived if
dL2β±k = −(∂−α
+k + ∂+α
−k + Fk(a1, · · · , ak−1, b1, · · · , bk−1)).
Next we prove that the right hand side above is dL2-closed, which allows us to concludethat the obstruction to find β±k lies in H(2,1)+(1,2).
We letb<k+1 =
∑j<k+1
1j!(β
±j + β±j)t
j .
Since e−aJ 1ea = e−ae−b<k+1J 1e
b<k+1ea is integrable for any choice of bk, we have that theoperator
e−b<k+1e−adHeaeb<k+1 |U0,n
has two components, say dH |Un,0 = ∂J 2ε +NJ 2ε,
∂J 2k = ∂J 2 + v(t), v(t) ∈ Γ(L2), v(0) = 0;
NJ 2ε = (NJ 2ε)ktk +O(tk+1) ∈ Γ(∧1,2L2).
According to Corollary 2.4, ∂J 2ε, NJ 2ε = 0, hence, in particular, the degree k part ofthis operator vanishes, i.e.,
0 = ∂J 2 , (NJ 2ε)k = dL2(NJ 2ε)k.
Remark. There are natural maps
π2,1 : H(2,1)+(1,2) −→ H2,1
∂+(M ;L2) and π1,2 : H(2,1)+(1,2) −→ H1,2
∂−(M ;L2).
given by taking the appropriate components of a class inH(2,1)+(1,2). It is possible, however,that a nontrivial class in H(2,1)+(1,2) is mapped into the trivial class by both projections.In terms of deformations, this means that there may be no obstruction to deforming ageneralized Kahler structure as either a positive or a negative SKT structure, while thegeneralized Kahler deformation problem itself is obstructed. Next theorem gives a concretesituation where this phenomenon happens.
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Theorem 8.9. Let (M,ω, I) be a compact Kahler manifold, ωt be a family of deforma-tions of the symplectic structure and J 1t be the corresponding family of generalized com-plex structures. Then, for t small, one can deform I into an generalized almost complexstructure, J 2t, such that (J 1t,J 2t) is a generalized complex extension of a positive SKTstructure.
Proof. Due to Moser’s trick, it is enough to consider linear deformations of ω by a harmonic2-form ε. Further if we let εp,q be the components of ε according to the usual (p, q)decomposition of forms determined by the complex structure, then ω + tε1,1 is still aKahler form with respect to I, so it is enough tho prove the result for a deformation givenby a harmonic form Re(ε2,0).
Next we observe that it is enough to prove the result for a deformation of the formω + tε0,2, with ε0,2 ∈ Ω0,2(M) a closed form. Indeed, let J 1t be the generalized complexstructure corresponding to the complex valued form ω + tε0,2, that is, J 1t has canonicalbundle ei(ω+tε0,2). If we can find a family of generalized almost complex structures J 2t
which agrees with J 2 at time zero such that (J 1t,J 2t) is a generalized complex extensionof an SKT structure, then by performing a B-field transform with B = tIm (ε0,2) we get ageneralized complex extension of an SKT structure with J 1t determined by ω+ tRe(ε0,2).
Since we have an isomorphism of Lie algebroids
L1 = X − iω(X) : X ∈ TCM ∼= TCM,
given by the projection onto TCM , we see that the differential complex (Γ(∧•L1), dL1) isisomorphic to (Ω•(M), d). In fact, the map ψ from Proposition 1.9 gives the isomorphism:
ψ : ∧•T ∗M −→ ∧•L1 ψ(ϕ) = eiωei2ω−1
ϕ.
And according to Proposition 5.14, the operators δ+, δ−, δ+ and δ− are related to ∂, ∂,
∂ω−1
and ∂ω−1
via Ψ:
Ψ−1δ+Ψ =i
2∂ω−1
; Ψ−1δ+Ψ = ∂;
Ψ−1δ−Ψ =i
2∂ω−1
; Ψ−1δ−Ψ = ∂.
Since ε ∈ Ω0,2(M), then ψ(ε) ∈ Γ(∧2L2) ∩ Γ(∧2L1) = Γ(∧2V 0,1− ) represents a dL2-
cohomology class which gives rise to the deformation of J 1 corresponding to the complexform ω + tε.
According to Proposition 8.5, there is a smooth family J 2t such that (J 1t,J 2t) is ageneralized complex extension of an SKT structure.
Remark. Of course, same result holds for negative SKT structures. That is any deforma-tion of the symplectic structure on a Kahler manifold can be completed to a generalizedHermitian structure which is an extension of either a positive or a negative SKT structure.
Notice that, in effect, this theorem considers a Kahler structure as a bihermitian struc-ture with I+ = I− and the crucial step is the deformation of the symplectic structureby a harmonic form ε ∈ Ω2,0(M). At that step, we use ε to deform I− into a (possibly)
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nonintegrable complex structure still compatible with the same metric for which J 1ε isthe desired symplectic structure. The SKT structure (g, I+) is just the original Kahlerstructure.
References
[1] M. Atiyah and G. Segal, Twisted K-theory and cohomology, Inspired by S. S. Chern,Nankai Tracts Math., vol. 11, World Sci. Publ., Hackensack, NJ, 2006, pp. 5–43.
[2] J.-M. Bismut, A local index theorem for non-Kahler manifolds, Math. Ann. 284(1989), no. 4, 681–699.
[3] H. Bursztyn, G. R. Cavalcanti, and M. Gualtieri, Reduction of Courant algebroids andgeneralized complex structures, Adv. Math. 211 (2007), no. 2, 726–765.
[4] H. Bursztyn, G. R. Cavalcanti, and M. Gualtieri, Generalized Kahler geometry ofinstanton moduli spaces, 2012, http://arxiv.org/abs/1203.2385.
[5] G. R. Cavalcanti, New aspects of the ddc-lemma, Ph.D. thesis, Oxford University,2004.
[6] G. R. Cavalcanti, The decomposition of forms and cohomology of generalized complexmanifolds, J. Geom. Phys. 57 (2006), no. 1, 121–132.
[7] G. R. Cavalcanti, Reduction of metric structures on Courant algebroids, J. SymplecticGeom. 4 (2006), no. 3, 317–343.
[8] P. de Bartolomeis, Symplectic deformations of Kahler manifolds, J. Symplectic Geom.3 (2005), no. 3, 341–355.
[9] B. de Wit and P. van Nieuwenhuizen, Rigidly and locally supersymmetric two-dimensional nonlinear σ-models with torsion, Nuclear Phys. B 312 (1989), no. 1,58–94.
[10] G. W. Delius, M. Rocek, A. Sevrin, and P. van Nieuwenhuizen, Supersymmetric sigmamodels with nonvanishing Nijenhuis tensor and their operator product expansion, Nu-clear Phys. B 324 (1989), no. 2, 523–531.
[11] N. Enrietti, A. Fino, and L. Vezzoni, Tamed Symplectic forms and SKT metrics, 2010,http://arxiv.org/abs/1002.3099.
[12] A. Fino, M. Parton, and S. Salamon, Families of strong KT structures in six dimen-sions, Comment. Math. Helv. 79 (2004), no. 2, 317–340.
[13] A. Fino and A. Tomassini, Blow-ups and resolutions of strong Kahler with torsionmetrics, Adv. Math. 221 (2009), no. 3, 914–935.
[14] S. J. Gates, Jr., C. M. Hull, and M. Rocek, Twisted multiplets and new supersymmetricnonlinear σ-models, Nuclear Phys. B 248 (1984), no. 1, 157–186.
[15] P. Gauduchon, Le theoreme de l’excentricite nulle, C. R. Acad. Sci. Paris Ser. A-B285 (1977), no. 5, A387–A390.
[16] R. Goto, Deformations of generalized complex and generalized Kahler structures, J.Differential Geom. 84 (2010), no. 3, 525–560.
63
[17] J. Grabowski, Courant-Nijenhuis tensors and generalized geometries, Groups, geome-try and physics, Monogr. Real Acad. Ci. Exact. Fıs.-Quım. Nat. Zaragoza, 29, Acad.Cienc. Exact. Fıs. Quım. Nat. Zaragoza, Zaragoza, 2006, pp. 101–112.
[18] G. Grantcharov, G. Papadopoulos, and Y. S. Poon, Reduction of HKT-structures, J.Math. Phys. 43 (2002), no. 7, 3766–3782.
[19] M. Gualtieri, Generalized geometry and the Hodge decomposition, 2004, http://
arxiv.org/pdf/math/0409093v1.
[20] M. Gualtieri, Generalized Kahler geometry, 2010, http://arxiv.org/pdf/1007.
3485v1.
[21] M. Gualtieri, Generalized complex geometry, Ann. of Math. (2) 174 (2011), no. 1,75–123.
[22] N. Hitchin, Brackets, forms and invariant functionals, Asian J. Math. 10 (2006),no. 3, 541–560.
[23] C. M. Hull and E. Witten, Supersymmetric sigma models and the heterotic string,Phys. Lett. B 160 (1985), no. 6, 398–402.
[24] K. Kodaira and D. C. Spencer, On deformations of complex analytic structures. III.Stability theorems for complex structures, Ann. of Math. (2) 71 (1960), 43–76.
[25] Y. Kosmann-Schwarzbach, Derived brackets, Lett. Math. Phys. 69 (2004), 61–87.
[26] J.-L. Koszul, Crochet de Schouten-Nijenhuis et cohomologie, Asterisque (1985),no. Numero Hors Serie, 257–271. The mathematical heritage of Elie Cartan (Lyon,1984).
[27] Z.-J. Liu, A. Weinstein, and P. Xu, Manin triples for Lie bialgebroids, J. DifferentialGeom. 45 (1997), no. 3, 547–574.
[28] M. Lubke and A. Teleman, The Kobayashi-Hitchin correspondence, World ScientificPublishing Co. Inc., River Edge, NJ, 1995.
[29] M. L. Michelsohn, Clifford and spinor cohomology of Kahler manifolds, Amer. J. Math.102 (1980), no. 6, 1083–1146.
[30] A. Neveu and J. H. Schwarz, Factorizable dual model of pions, Nuclear Physics B 31(1971), no. 1, 86 – 112.
[31] J. Wess and B. Zumino, Consequences of anomalous Ward identities, Phys. Lett. 37B(1971), 95–97.
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