Advances in Mathematics

We use tools from generalized complex geometry to develop the theory of SKT (a.k.a. pluriclosed Hermitian) manifolds and more generally manifolds with special holonomy with respect to a metric connection with closed skew-symmetric torsion. We develop Hodge theory on such manifolds showing how the reduction of the holonomy group causes a decomposition of the twisted cohomology. For SKT manifolds this decomposition is accompanied by an identity between diﬀerent Laplacian operators and equates diﬀerent cohomologies de-ﬁned in terms of the SKT structure. We illustrate our theory with examples based on Calabi–Eckmann manifolds, instantons, Hopf surfaces and Lie groups.


Introduction
Looking beyond the Levi-Civita connection in Riemannian geometry, one finds a number of other metric connections with interesting properties.Normally these families of connections are defined by some characteristic of the torsion tensor and recurrent themes of research are connections with parallel torsion or connections with skew symmetric torsion.The "strong" torsion condition refers to the latter: a strong torsion connection on a Riemannian manifold is a metric connection whose torsion is skew symmetric and closed.In this setting, a Kähler structure with strong torsion, or SKT structure is a Hermitian structure (g, I) together with a strong torsion connection for which I is parallel.A weaker notion is that of a (strong) parallel Hermitian structure which for us means 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 that in the former case integrability of the complex structure is not required.
A reason to study of such objects comes from string theory, where closed 3-forms arise naturally as fields in their sigma models [22,26].Once a 3-form is added to the sigma model, if one still requires a nontrivial amount of supersymmetry, the type of geometry of the target space has to move away from the usual Kähler geometry.It was precisely following this path that Gates, Hull and Roček [12] discovered the bi-Hermitian geometry that nowadays also goes by the name of generalized Kähler geometry [16] as the solutions to the (2, 2)-supersymmetric sigma model.Requiring less supersymmetry without giving up on the idea altogether leads one to consider models where there is more left than right supersymmetry or models where the right side is simply absent.These conditions lead to (2,1) or (2, 0) supersymmetric sigma models and supersymmetry holds if and only if the target space has an SKT structure [20].This point of view also leads one to consider parallel Hermitian and bi-Hermitian structures as these are geometric structures imposed by a sigma model with an extended supersymmetry algebra [6,7].
Mathematical properties of SKT structures have been subject of study of several papers since the 90's, including those of Bismut [1], Grantcharov et al. [14], Fino and collaborators [9,10] and more recently Streets and Tian [23,24].Yet, until now there was no framework in which one could extend results regarding Hodge theory of Kähler manifolds to the SKT world.Indeed the opposite seemed to be the case: Since SKT manifolds are, in particular, complex, their space of forms inherits a natural bi-grading, but a simple check in concrete examples shows that there is no corresponding decomposition of their cohomology.Further, by studying these structures on six dimensional nilmanifolds Fino, Parton and Salamon [9] produced examples showing that their Frölicher spectral sequence does not necessarily degenerate at the first page, they do not satisfy the dd clemma, manifolds carrying these structures may not be formal and that these structures are not stable under deformations of the complex structure.In short, several Kähler properties seem to have been lost once the torsion was included.Put another way, given a complex manifold, we were left with no tools to decide whether it admitted an SKT metric or not.
Here we tackle SKT structures from a new point of view.The key observation is that SKT structures have yet another description, this time, as a 'generalized structure', i.e., a geometric structure on T M = T M ⊕ T * M .In fact, in this paper we show that, in a very precise way, SKT structures lie half way between generalized Hermitian and generalized Kähler structures.Using this approach, we show that some of the negative results mentioned earlier have a positive counterpart involving the torsion while those with negative answers obtain a conceptual explanation for their failure to hold.
Indeed, the first observation is that, as structures defined on T M with the H-Courant bracket, the natural differential operator to consider is d H = d + H∧, where H = d c ω, and ω is the Hermitian form.Hence questions about d, and its decomposition as ∂ + ∂ miss an important ingredient and were doomed from the start.The cohomology of d H is only Z 2 -graded, yet we show that a parallel Hermitian structure induces a Z ×Z 2 grading on the space of forms which itself induces a Z ×Z 2 grading on the d H -cohomology.This is achieved by introducing the intrinsic torsion of a generalized almost Hermitian structure and using it to show that the d H -Laplacian preserves a Z × Z 2 -graded decomposition of the space of forms.For SKT manifolds one can go further and prove an identity of Laplacians, extending Gualtieri's work on generalized Kähler geometry [15].This way we relate a cohomology naturally defined in terms of the SKT data with the d H -cohomology.As an application, we return to the moduli space of instantons on a bundle over a compact complex surface and show that the existence of an SKT structure in this space (obtained by Lübke and Teleman [21] by an ad hoc method) can be seen as a consequence of the Hodge theory developed for SKT manifolds.
This paper is organized as follows.In Section 1 we develop the linear algebra pertinent to generalized complex, generalized Hermitian and SKT structures.In particular we show that an SKT structure gives rise to a Z × Z 2 -grading on the space of forms.In Section 2 we introduce the intrinsic torsion of a generalized Hermitian structure and, in Sections 3 and 4, we relate SKT structures and parallel Hermitian structures to the vanishing of certain components of the intrinsic torsion.In Section 5 we study Hodge theory for SKT and parallel Hermitian structures and prove that in both cases the d H -cohomology decomposes according to the decomposition of forms induced by the structure.As an application, in Section 7 we recover the result that the moduli space of instantons over a complex surface has an SKT structure.
Acknowledgments: This research was supported by the Marie Curie Intra European Fellowship PIEF-GA-2008-220178 and the VIDI grant 639.032.221from NWO, the Dutch science foundation.The author is thankful to Anna Fino, Sönke Rollenske, Ulf Lindstrom, Martin Roček, Stefan Vandoren and Maxim Zabzine for useful conversations.

Linear algebra
Given a vector space V m we let V = V ⊕ V * be its "double".V is endowed with a natural symmetric pairing: hence ∧ • V * is naturally a module for the Clifford algebra of V .In fact, it is the space of spinors for Spin(V ) and hence comes equipped with a spin invariant pairing, the Chevalley pairing: where • t indicates transposition, an R-linear operator defined on decomposable forms by 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 and on ∧ • V * by the Clifford action described above, so we have Exponentiating this map we have And this action is compatible with the action of e B ∈ Spin(V ) on forms: So we have e B * : We will be interested in introducing geometric structures on V .The first we consider is a generalized metric, as introduced by Gualtieri [16].Definition 1.2.A generalized metric on V is an automorphism G : V −→ V which is orthogonal and self-adjoint with respect to the natural pairing and for which the bilinear tensor Since G is orthogonal and self-adjoint, we have Since both V and GV are isotropic subspaces of V which project isomorphically onto V , we can describe GV as the graph of a linear map b : Isotropy means that b ∈ ∧ 2 V * and hence gives rise to an orthogonal transformation of the natural pairing, e b * .This map has the property that e b * : GV * −→ V , hence, after an orthogonal transformation of V , we can assume that GV * = V .For this splitting, where g is an ordinary metric on V .The splitting of V determined by a generalized metric is the metric splitting.
If V is endowed with an orientation, we can define a generalized Hodge star operator [15] as follows.Since π With this definition, we have If the splitting of V is the metric splitting, we have where * is the usual Hodge star, hence, in this splitting, is the usual Hodge star except for a change in signs given by the Chevalley pairing.Since = −e m • . . .• e 2 • e 1 , we have that 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.This allows us to define self-dual and anti self-dual forms in all dimensions: We denote the space of SD-forms by ∧ • + V * and the space of ASD-forms by The Clifford action of elements in V ± either preserves or switches the eigenspaces of : Then acting via Clifford action on forms we have hence for m even, and for m odd we can choose an orthonormal basis for V + for which e 1 = v/ v , so that v anti commutes with all the remaining elements of the basis and commutes with e 1 .
For the rest of this section we will introduce structures on V which force its dimension to be even so we let m = 2n.Definition 1.5.A generalized complex structure on V is a complex structure on V which is orthogonal with respect to the natural pairing.A generalized Hermitian structure or a U (n) ×U (n) structure on V is a generalized complex structure J 1 on V and a generalized metric G such that J 1 and G commute.
Given a generalized complex structure J on V , we can split V C , the complexification of V , into the ±i-eigenspaces of J : V C = L ⊕L.The spaces L and L are maximal isotropic subspaces of V C such that L ∩ L = {0}.Since the natural pairing is nondegenerate, we can use it to identify L = L * .Precisely, we identify Given a generalized Hermitian structure (J 1 , G) on V , J 2 = GJ 1 is orthogonal with respect to the natural pairing and squares to −Id, hence it is also a generalized complex structure.Since π V : V ± −→ V are isomorphisms, and J 1 | V ± is a complex structure on V ± orthogonal with respect to the natural pairing, it induces complex structures I ± on V compatible 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 the eigenspaces of J 1 and J 2 : In this context, the generalized Hodge star is related to the action of 2 , namely: Lemma 1.6.(Gualtieri [15]) In a generalized Hermitian vector space one has This means that we can read the decomposition of forms into SD and ASD from the U p,q decomposition, namely | U p,q = i p+q .If we plot the (nontrivial) spaces U p,q in a lattice, each diagonal is made either of SD-or ASD-forms, with U n,0 made of SD-forms (see Fig. 1).Definition 1.7.A positive U (n) structure or a positive Hermitian structure on V is a generalized metric G and a complex structure Fig. 1.Representation of the spaces of SD and ASD in terms of the (p, q)-decomposition of forms on a 4-dimensional generalized Hermitian structure.
with respect to the natural pairing.A negative U (n) structure or negative Hermitian structure 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/negative U (n) structure (G, I) if I is the restriction of J to the appropriate space and (G, J ) is a generalized Hermitian structure.
Given a generalized Hermitian structure, since J 1 and G commute, J 1 preserves the eigenspaces of G and hence, upon restriction to V ± , one obtains a positive and a negative Hermitian structure.Conversely, a positive (resp.negative) Hermitian structure can be extended to V by declaring that it vanishes on V − (resp.V + ).Then a pair of positive and negative Hermitian structures, I + , I − gives rise to a generalized Hermitian structure by declaring 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 vector space (V, g, I).Further, we can use I + to define a complex structure I − on V − using the isomorphisms V + ∼ = V ∼ = V − and this way we have an extension of the U (n) structure to a generalized 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 the complex structure2 I and consequently J 2 is the generalized complex structure associated to the Hermitian form ω = g • I: For this set of choices, there is a relation between the (p, q)-decomposition of forms determined by the generalized Hermitian structure and the usual (p, q)-decomposition of forms determined by the complex structure I on V .Proposition 1.8.(Cavalcanti [4]) Let J 1 and J 2 be as above and let L 2 be the +i-eigenspace of J 2 .The map preserves J 1 and Therefore the corresponding action on spinors, preserves the eigenspaces of J 1 and maps A positive Hermitian structure is fully determined by the +i-eigenspace of + \{0}.If we are given a generalized complex extension J of a positive Hermitian structure (G, I + ) we obtain a bigrading of forms into U p,q as explained earlier.However, from the point of view of the U (n) structure, the natural spaces to consider are (1.7) Indeed W n + corresponds to the vector space of all forms that are annihilated by the Clifford action of V 1,0 + and W n−2k that is, the spaces W k + are solely determined by the complex structure I + .Notice 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 U p,q decomposition, each W k + is made of either SD-or ASD-forms, with W n + being SD.Another description of the spaces W • + is obtained by extending I + to an endomorphism of T M by declaring that I + | V − vanishes, so that I + is a skew-symmetric operator on T M with respect to the natural pairing, that is, I + ∈ spin(T M ).Similarly to a generalized complex structure, J , for which the space U k is the ik-eigenspace of the action of J , letting I + act on forms one sees that the space W k + is the i k 2 -eigenspace of I + .Hence, Lemma 1.6 takes the following form for positive U (n) structures: Lemma 1.9.Let (G, I + ) be a positive Hermitian structure on V , and let I + = e πI + 2 .Then Similarly, for a negative Hermitian structure (G, I − ), we can extend I − to V by declaring that it vanishes on V + .If J 1 is a generalized complex extension of I − , then the eigenspaces of I − correspond to the anti-diagonals of the U p,q decomposition: W k − = 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 × Z 2 -grading: In what follows we will refer to both spaces W k ± and W k,l ± , with the understanding that if the Z 2 -grading is not particularly important, we will simply omit it.

Intrinsic torsion of generalized Hermitian structures
Except for a generalized metric, each of the structures introduced in Section 1 has an appropriate integrability condition.We let (M 2n , H) be a manifold with a real closed 3-form H and consider the Courant bracket on sections of T M = T M ⊕ T * M : 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 d H = d + H∧, i.e., the following identity holds for all v 1 , v 2 ∈ Γ(T M ) and ϕ ∈ Ω • (M ): where • denotes the Clifford action of Clif(T M ) on ∧ • T * M and {•, •} denotes the graded commutator of operators.
The orthogonal action of a 2-form B ∈ Ω 2 (M ) on T M relates different Courant brackets: Definition 2.1.For each of the structures introduced in Definitions 1.5 and 1.7, we refer to the smooth assignment of such structure to T x M for each x ∈ M by including the adjective almost in the name of the structure.

Definition 2.2 (Integrability conditions).
• A (integrable) 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 compatible integrable generalized complex structure.• A generalized Kähler structure is a generalized Hermitian structure (G, J 1 ) for which J 1 and J 2 = J 1 G are integrable.

The Nijenhuis tensor
Let us spend some time to understand the Nijenhuis tensor of an almost generalized complex structure J .This tensor is defined in the usual way, namely if L is the +ieigenspace of J where • L indicates projection onto L. We can alternatively use the identification L = L * from (1.4) to consider the operator Further, the tensor N ∈ ∧ 2 L ⊗L defined above actually lies in ∧ 3 L. Indeed, for X, Y, Z ∈ Γ(L) we have which shows that N is fully skew.
A different way to understand N arises by using the U k decomposition of forms determined by J .Namely, letting U k = Γ(U k ) (throughout the paper we denote the sheaf of sections of U k by U k and the sheaf of sections of U p,q by U p,q ), one has: (2.4) Proof.We prove first that and will do so by induction, starting at k = n + 1 and working our way down.For k = n + 1 we have that U n+1 = {0} and the claim follows trivially.
Next we assume the result to be true for all j > k and let ρ ∈ U k .For v 1 , v 2 ∈ Γ(L), using (2.1) we have Since the Clifford action of v i sends U j to U j+1 , the inductive hypothesis implies that the last three terms lie in ⊕ j≥k−1 U j , while the first term, being the action of an element of L ⊕ L on ρ lies in U k+1 ⊕ U k−1 .Therefore we conclude that and (2.5) follows.
As d H is a real operator and U −k = U k conjugating (2.5) we have that Furthermore, if U k is made of even forms then U k+1 is made of odd forms and vice versa, we have that therefore proving (2.4).Now we prove that π k+3 • d H corresponds to the Clifford action of N .Once again we use 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 where in third equality we have used that the component of v 1 , v 2 ρ in U k+1 is given by the Clifford action of the L component of v 1 , v 2 , that is ι v 2 ι v 1 N and in the fourth equality we used the inductive hypothesis as well as the fact that when acting on forms, the interior product of v ∈ Γ(L) with ϕ ∈ Γ(∧ • L) is given by ϕ(v)ρ = {v, ϕ}ρ.The last equality follows by expanding the graded commutator and canceling out similar terms.
If we compose d H | U k with projection onto U k+1 and U k−1 we get operators and if the generalized complex structure J is clear from the context, we denote these operators simply by ∂ and ∂.As proved by Gualtieri in [17], integrability is equivalent to the requirement so we can also see from this point of view how the vanishing of the Nijenhuis tensor implies integrability.

The intrinsic torsion and the road to integrability
With this understanding of the Nijenhuis tensor, we can give a pictorial description of the long road to integrability from almost generalized Hermitian to generalized Kähler.Indeed, given an almost generalized Hermitian structure, we get a splitting of forms into spaces U p,q .According to Lemma 2.3, d H can not change either the 'p' or the 'q' grading by more than three and it must switch parity.Hence d H decomposes as a sum of eight operators and their complex conjugates and we can draw in a diagram all the possible nontrivial components of d H | U p,q as arrows (see Fig. 2).
Once we require that J 1 is integrable, i.e. we are dealing in fact with a generalized Hermitian structure, then d H only changes the 'p' degree by ±1 and several components of d H present in the nonintegrable case, now vanish and the diagram from Fig. 2 clears up to the one presented in Fig. 3.
Finally, if we require that J 2 is also integrable, and hence we are in fact dealing with a generalized Kähler structure, the last two components of the Nijenhuis tensor, labeled N 1 and N 2 above, vanish and d H decomposes as a sum of four operators, as pictured in Fig. 4.
Fig. 2. Representation of the nontrivial components of d H when restricted to U p,q for an generalized almost Hermitian structure.This shows that the obstruction for a generalized almost Hermitian structure to be a generalized Kähler structure is given by the tensors N i , i = 1, 2, 3, 4 and N ± Definition 2.4.The intrinsic torsion of a generalized Hermitian manifold are the tensors N i , i = 1, 2, 3, 4 and N ± .
In the next sections we will introduce geometric structures on M which are weaker than generalized Kähler structures and show how these structures can be phrased in terms of the vanishing of certain components of the intrinsic torsion.

The operators δ ± and δ ±
Not requiring integrability, d H restricted to U p,q has sixteen components but only four are not tensorial, i.e., linear over C ∞ (M ), namely δ ± and δ ± .Hence we have where ∼ indicates that these operators agree up to lower order terms, i.e., they have the same symbol.Using the decomposition one can easily see that the decomposition (2.7) in terms of symbols corresponds to the decomposition of a 1-form ξ ∈ T M into its four components according to (2.8) and one can also check that the symbol sequence for each of the operators δ ± and δ ± is exact, e.g., for δ + , the symbol sequence associated to sequence of operators is an exact sequence.Adding over p + q = k and letting W k + = ⊕ p+q=k U p,q (i.e., W k + is the sheaf of sections of the bundles W k + introduced in (1.7)) we have also that the symbol sequence associated to is exact.And similarly, adding over q we get that the symbol sequence of is exact.

Parallel Hermitian and bi-Hermitian structures
The first type of structures that we will relate to the intrinsic torsion are Hermitian structures which are parallel for a connection with closed skew torsion.The existence of a relationship between connections with closed, skew symmetric torsion and the Courant bracket was made evident by Hitchin in [19].Precisely, given a generalized metric on T M , we let H be the 3-form corresponding to the metric splitting and g be the induced metric on M .Also, for X ∈ Γ(T M) we let X ± ∈ Γ(V ± ) be unique lifts of X to V ± , we let π ± : T M −→ V ± be the orthogonal projections onto V ± and π T : T M −→ T M be the natural projection.Proposition 3.1 (Hitchin [19]).Let ∇ ± be the unique metric connection whose torsion is skew symmetric and equal to ∓H.Then From this proposition, we see that the isomorphisms π T : V ± −→ T M relate the connections with torsion ∓H to the operators As we will work with the spaces V ± directly, we will use ∇ ± instead of the connections they induce on T M, with the understanding that these are equivalent operators.Therefore, for example, if ∇ + has holonomy in U (n), then M has an almost Hermitian structure (g, I) which is parallel with respect to ∇ + and the isomorphism π T : And conversely, a Hermitian structure satisfying (3.3) is parallel with respect to ∇ + .The same result holds exchanging Proof.Let I + be a complex structure on V + orthogonal with respect to the natural pairing and let V 1,0 + be its +i-eigenspace.Then, for v 1 , v 2 ∈ Γ(V 1,0 + ) and w ∈ Γ(V − ), we have that v i , w = 0, as V + and V − are orthogonal with respect to the natural pairing.Hence where in the third equality we used again that V + and V − are orthogonal with respect to the natural pairing, and hence v 1 , w = − w, v 1 and in the fourth equality we used that v 2 ∈ V + and hence the V − component of w, v 1 is annihilated by the natural pairing.
) and w ∈ Γ(V − ) and hence, according to (3.4), so does w, v 1 , v 2 , showing that v 1 , v 2 must be orthogonal to V − and hence, a section of Next, and throughout these notes, we denote by W k and W k,l the sheaf of sections of the bundles W k and W k,l respectively.Theorem 3.4.Let (M, H) be a manifold with 3-form, G be a generalized metric on M and I + , I − be a positive and a negative almost Hermitian structure on M .The following hold: Remark.In comparison with Lemma 2.3, the indices of the spaces W • ± might seem a little strange, but one should keep in mind that if Proof.Since V − is the orthogonal complement of V + with respect to the natural pairing, it is clear that (3.3) is equivalent to the following two conditions but the first condition is equivalent to the vanishing of N 2 and the second, to the vanishing of N 3 .Finally, since the component of d H mapping W k to W k+4 is given by the sum N 2 + N 3 , we see that the vanishing of this component is also equivalent to the parallel condition.
The remaining claims are proved similarly.The decomposition of d H for an almost generalized Hermitian structure extending a parallel positive Hermitian structure is depicted in Fig. 5 and the decomposition of d H for a parallel bi-Hermitian structure is depicted in Fig. 6.

SKT structures
Classically, an SKT structure is a Hermitian structure (M, g, I) for which dd c ω = 0, where ω is the Hermitian 2-form.In this case, d c ω = H is a closed 3-form and the complex structure is in fact parallel with respect to the metric connection with torsion −H, hence SKT structures are a particular case of positive parallel Hermitian structures, the only difference being that now one requires I to be integrable.In this section we phrase the SKT condition in terms of generalized geometry and relate it with the intrinsic torsion.We will see that an SKT structure lies precisely half way between a generalized Hermitian and a generalized Kähler structure.
Given an SKT structure (g, I) we let H be the background 3-form and consider the generalized metric G as in (1.2).Using the isomorphism V + ∼ = T M we use I to induce a complex structure I + on V + and hence split V + into eigenspaces of Proposition 4.1.(Cavalcanti [5]) For an SKT structure (g, I) on M with Conversely, given a positive almost Hermitian structure (G, I + ) on (M, H), where H is the 3-form associated of the metric splitting of T M , if V 1,0 + is involutive, then the induced Hermitian structure (g, I) on M is an SKT structure with d c ω = H.
Remark.Here we obtain, in a new light, a well known contrast between connections with torsion and the Levi-Civita connection regarding integrability.Indeed, for the Levi-Civita connection, reduction of the holonomy group to U (n) implies integrability of the complex structure, but that is known not to be the case for connections with torsion.From our point of view, this is the difference between the reduced holonomy condition and the SKT condition Of course, if we have d c ω = −H, we can lift the SKT data to V − to obtain a negative Hermitian structure and an analogous version of Proposition 4.1 holds.
This result motivates our formulation of the SKT condition.
Definition 4.2.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 fiberwise an extension of I, in which case we say that J is a generalized (almost) complex/Hermitian extension of the SKT structure.
Firstly, we observe that the SKT condition can also be phrased in terms of the vanishing of components of the intrinsic torsion.Indeed, in the presence of an extension of, say, a positive 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 that is, the components N 2 , N 3 and N + of the intrinsic torsion vanish.A similar argument gives a characterization of negative SKT structures.
Corollary 4.5.Let M 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 Kähler structure.
Proof.Since M is four dimensional, V 1,0 ± are two dimensional complex vector spaces, so ± are the trivial tensors.
Corollary 4.6 (Gualtieri [16]).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 and ) is a generalized Kähler structure.
Proof.According to Proposition 4.3, under the hypothesis, all components of the intrinsic torsion vanish hence (G, J 1 ) is a generalized Kähler structure.
Next, we describe the integrability condition for an SKT structure in terms of the decomposition of forms into W k ± and U p,q (for a fixed generalized complex extension J 1 ) described in the previous section.Theorem 4.7.Let (G, I + ) be a positive almost Hermitian structure on a manifold with 3-form (M 2n , H).Then the following are equivalent: Proof.We will first prove that 1) implies 2).Let I − be any complex structure on V − orthogonal with respect to the natural pairing so that G and J 1 = I + + I − form a generalized almost Hermitian structure.Then according to Proposition 4.3, N 2 , N 3 and N + vanish and hence 2) holds.Condition 2) clearly implies condition 3).Finally to prove that 3) implies 1) we once again choose a complex structure I − on V − and observe that since N 2 , N 3 and N + are tensors, it is enough to check that they vanish when applied to spaces where their action is effective.But for ϕ ∈ U n,0 ⊂ W n + , the components of d H landing on W n−6 + and W n−4 + are precisely N + • ϕ and N 3 • ϕ and the vanishing of these forms for ϕ = 0 implies that hence the vanishing of this component implies that N 2 = 0 and Proposition 4.3 implies that I + is integrable.
Definition 4.8.Let (M, G, I + ) be a positive SKT structure.We define as the different components of d H according to the decomposition from Theorem 4.7 so While the different U p,q components of d H obtained in terms of the generalized almost Hermitian extension of the SKT structure depend on the particular extension chosen, the operators δ N + , δ N + and / δ − depend only on the SKT data, since they are the decomposition of d H obtained from the eigenspaces of I + (Fig. 7).Since (d H ) 2 = 0 these operators satisfy some relations.
Corollary 4.9.The following hold In the arguments up to now, given a positive SKT structure, we have chosen I − rather freely, but the complex structure I corresponding to the SKT data (see Proposition 4.1) is integrable and together with G forms a generalized Hermitian extension of (G, I + ), that is, an SKT always has a generalized Hermitian extension.Theorem 4.10.Let (M, H) be a manifold with 3-form and (G, J 1 ) be an generalized almost Hermitian structure on M .Then the following are equivalent: 1. (G, J 1 ) is an integrable generalized complex extension of a positive SKT structure; Proof.The fact that 1) implies 2) follows from Proposition 4.3 and integrability of J 1 .The implications 2) ⇒ 3) ⇒ 4) are immediate.Finally, similarly to the proof of Theorem 4.7, the values of the different components of the intrinsic torsion are fully determined by their action on U n,0 and U 0,n and 4) implies the vanishing of all components of the intrinsic torsion except from N 1 .
This theorem allows us to define six differential operators on a generalized complex extension of a positive SKT structure (Fig. 8): We see that N = N 1 is the Nijenhuis tensor of J 2 and this theorem shows that if (G, J 1 ) is a generalized Hermitian extension of an SKT structure then half of the Nijenhuis tensor of J 2 vanishes, namely the component N 2 from (2.6).Also, this decomposition allows us to express the operators δ N + and / δ − from (4.1) in terms of δ + , δ − , N and their conjugates: Since (d H ) 2 = 0 these operators satisfy a number of relations.
Corollary 4.11.The following relations and their complex conjugates hold: Corollary 4.12.Let (G, J 1 ) be a generalized Hermitian structure on a manifold with 3form (M, H) and let J 2 = GJ 1 be the associated generalized almost complex structure.If the canonical bundle of J 2 admits a ∂ J 1 -closed trivialization, then (G, J ) is an extension of an SKT structure. Proof.
Due to Lemma 2.3, we have that But according to the hypothesis there is a trivialization ρ of U 0,n such that ∂ J 1 ρ = 0, so, in particular, N 2 • ρ, the component of ∂ J 1 ρ in U −1,n−3 , must vanish and since it vanishes on ρ, N 2 is the zero tensor.So we have in fact and the last condition of Theorem (4.10) holds.
The same results hold for negative SKT structures.One should bear in mind, however that the relevant spaces for a negative structure are given by W k − = p−q=k U p,q , that is, each W k − is an antidiagonal and integrability is equivalent to or, in terms of the U p,q decomposition obtained by choosing a generalized complex extension, the only nontrivial component the Nijenhuis tensor is N 2 .So an SKT structure (positive or negative) corresponds to a generalized Hermitian structure in which half of the Nijenhuis tensor of J 2 vanishes.

Hodge theory
In this section we develop Hodge theory for manifolds with parallel Hermitian, bi-Hermitian and SKT structures.Our main result is that for a parallel positive Hermitian structure the Laplacian preserves the spaces W k,l + and hence induces a decomposition of the d H -cohomology accordingly.This is in contrast with the fact that for usual manifolds the d H -cohomology has only a Z 2 -grading.For positive SKT manifolds, not only does the Laplacian preserve the spaces W k,l + , but in fact there is an identity between the Laplacians for the operators d H , δ N + and δ N + .We start with real Hodge theory and some operators of interest.

Differential operators, their adjoints and Laplacians
Given a generalized metric and orientation on a compact manifold M m , we can form the Hodge star operator which gives us a positive definite inner product on forms: Two basic results about d H are: (Integration by parts) hence the formal adjoint of d H is given by d H * = (−1) We will be mostly interested in the Dirac operators:

The operators / D
H ± relate to the projections of d H onto SD-and ASD-forms

In particular we see that / D
H − preserves the spaces Ω • ± (M ) while / D H + maps them to each other.

Proof.
/ D Therefore H preserves the decomposition of forms into Ω • ± (M ) and hence the d Hcohomology of M splits as SD-and ASD-cohomology:

Not only does / D
H + send even forms to odd and vice versa but, according to Lemma 5.2, it swaps Ω

Signature and Euler characteristic of almost Hermitian manifolds
In a compact generalized almost Hermitian manifold we have sixteen operators induced by the U p,q decomposition of forms.It turns out that for these operators taking complex conjugates or adjoints are nearly the same thing.Firstly, we extend the real inner product on forms (5.1) to complex valued forms by requiring it to be Hermitian.If we let denote the operator given by ϕ = ϕ, we have where the sign for the adjoint is positive if Proof.The proof that integration by parts holds is the same for all of these operators, so we consider only δ + .It is enough to consider the case when ϕ ∈ U p,q and hence δ + ϕ ∈ U p+1,q+1 and it pairs trivially with U k,l , unless k = −p − 1 and l = −q − 1. Hence we may assume that ψ ∈ U −p−1,−q−1 and compute where in the first equality we have used that the remaining components of d H ϕ do not lie in U p+1,q+1 , hence they pair trivially with ψ, in the second equality we integrated by parts and then reversed the argument.
For the formal adjoint, again taking δ = δ + , ϕ ∈ U p,q and ψ ∈ U p+1,q+1 we compute: where we have used several times that on U p,q is multiplication by i p+q and in the last equality we used that p + q = n mod 2.
Therefore we can form the Dirac operator corresponding to, say, δ + : Since the symbol sequence of δ + is exact, we get elliptic operators for the sequences (2.9) and (2.10): The indices of these operators are just the Euler characteristic and the signature of M : Theorem 5.6.Let (M, H) be a compact manifold with a 3-form H and let (G, J 1 ) be an almost generalized Hermitian structure on M .Let h ev/od = dim(ker(/ δ + : Ω ev/od −→ Ω od/ev )), Proof.To prove the claim about the Euler characteristic we only have to show that the symbols of / δ + and / D H + : Ω ev −→ Ω od since due to Lemma 5.4 the index of / D H + is the Euler characteristic.To compare the symbols we have where ∼ means that the operators have the same symbol and in the second passage we used Proposition 5.5.
The proof of the claim regarding the signature is done along the same lines.Proof.The claim for positive and negative structures are analogous and together they imply 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 of the U p,q decomposition lies in either Ω

Hodge theory on parallel Hermitian manifolds
If I + is parallel, then, according to Theorem 3.4, N 2 = N 3 = 0 and from (5.2) we have that / D H − ) 2 also preserves the W k decomposition of forms.Since the Laplacian has even parity, it also preserves Ω ev (M ) and Ω od (M ), hence the Laplacian preserves the spaces W k,l + .
• An SKT structure on M induces a Z × Z 2 -grading on the d H -cohomology; • (Gualtieri [15]) A generalized Kähler structure induces a Z 2 -grading on the d Hcohomology.
For each of the cases covered in the previous theorem, we denote by H •,• d H (M ) the corresponding decomposition of the d H -cohomology.
Corollary 5.9.(Riemann bilinear relations) In a compact parallel positive Hermitian manifold Proof.Indeed, let α be the harmonic representative for the class Finally in this context we get a clearer version of Theorem 5.6: Corollary 5.10.Letting w k,l be the dimension of the space of harmonic sections of W k,l + in a parallel Hermitian manifold we have While for parallel Hermitian manifolds we could prove that the Laplacian preserves the spaces W k,l + , in and SKT manifold we can go further and show that there is an identity of Laplacians: Theorem 5.11.In a positive SKT manifold, 3 and Proposition 5.5 we have

Relation to Dolbeault cohomology
As we saw in Proposition 4.1, given a positive SKT structure (G, I) on M one can extend I + to a generalized complex structure J 1 of complex type.In this case, in the metric splitting of T M , the structures J 1 and J 2 = GJ 1 are given by where ω = g • I and for such generalized Hermitian structure Proposition 1.8 gives an automorphism of the space of forms which relates the U p,q decomposition of forms with the usual ∧ p,q T * M decomposition of forms determined by the complex structure I.
In this section we relate these two decompositions of forms and corresponding cohomologies.Precisely, the effect of applying the automorphism Ψ from Proposition 1.8 to Since d H splits according to the U p,q into six operators, the same is true for dH and we can define Proposition 5.12.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 4.1.Let Proof.The proof is a direct computation of the operator dH : We compute separately each of the two operators above making up dH : The first term is just d = ∂ + ∂, while the second term is a version of the symplectic adjoint of d now obtained in a nonintegrable setting, Since ω is not closed, δ ω −1 does not square to zero.Integrability of the complex structure gives d = ∂ + ∂ and recalling that ω −1 is of type (1, 1) we have Since ω −1 is even, the third term in the series coincides with a multiple of the expression for the derived bracket of ω −1 with itself, that is, the Schouten-Nijenhuis bracket: Since ω −1 is of type (1, 1), integrability of the complex structure implies that this term this is a 3-vector lies in ∧ 2,1 T ⊕ ∧ 1,2 T , so this term decomposes as which vanishes since ω −1 is a bivector and hence so do the remaining terms in the series.So we have established that Next we compute the second summand in (5.4): where we have used (1.1) in the second equality.The element e 2 ω −1 (ξ) + ξ, we see that the 3-form component is ∂ω and that the 3-vector component is − i 8 ω −1 ∂ω.We let ζ 1 be the component of e Putting this together with (5.5) and the fact that dH does not have a component mapping Ω p,q (M ) to Ω p−2,q−1 (M ) we conclude that χ = −2ω −1 (∂ω) and obtain (5.3).
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 and the parity of the anti-holomorphic degree, i.e., ∂ i∂ω : Ω p,q (M ) −→ Ω p+1,q (M ), p ∈ Z, q ∈ Z 2 . (5.6) Therefore its cohomology has a natural Z × Z 2 -grading.For an operator δ with δ 2 = 0 we denote its cohomology by H δ (M ).
Corollary 5.13.Let (G, I + ) be a positive SKT structure on a compact manifold, let J be the generalized complex extension from Proposition 4.1 and let (g, I) be the corresponding Hermitian structure with Hermitian form ω = g • I. Then for p, q ∈ Z Proof.Indeed, Ψ puts Ω p,q (M ) in correspondence with U q−p,n−p−q and Ω p,• (M ) with W n−2p and according to Proposition 5.12, Ψ Of course the cohomology of ∂ 2i∂ω is the same as the cohomology if ∂ i∂ω , as the automorphism for all α ∈ ∧ p,q T * M, relates them.
The isomorphism Ψ used to prove Proposition 5.12 can be used even in the nonintegrable case to give us information about the Euler characteristic and signature of almost Hermitian manifolds.Indeed, according to Proposition 1.8 we have And hence conjugating δ + by Ψ we get an operator Since the isomorphism Ψ from Proposition 1.8 identifies T * 1,0 M with V Then In particular in a compact complex manifold if h p,q = dim(H p,q ∂ (M )), then χ = (−1) p+q h p,q and σ = p,q (−1) q h p,q (M ). (5.7) Remark.Using Frölicher's spectral sequence, Frölicher proved the identity for the Euler characteristic assuming integrability of the complex structure [11].The identity regarding the signature is an extension of the Hodge Index Theorem, which, in its modern version (see, e.g., [25], Theorem 6.33), states that (5.7) holds on a compact Kähler manifold and is obtained as a consequence of the development of Hodge theory of Kähler manifolds.
The main point of Corollary 5.14 is that these identities do not depend on existence of Kähler structures or on the integrability of the complex structure, so, in effect, they do not represent an obstruction to the existence of any of the structures studied here.This is to be compared with Corollary 5.13 which provide a nontrivial differential-topological obstruction to the existence of SKT structures on complex manifolds.
Quite separate from the theory developed so far, one can get other obstructions to the existence of SKT structures using only classical tools.For example: Theorem 5.15.A compact SKT manifold (M, I, g) for which admits a symplectic structure.
Example 5.16 (Calabi-Eckman manifolds).Among the Calabi-Eckman manifolds, S 1 × S 1 , S 1 × S 3 and S 3 × S 3 are known to admit SKT structures, by virtue of being compact Lie groups.In this example we show that all the remaining Calabi-Eckman manifolds M u,v ∼ = S 2u+1 × S 2v+1 do not admit SKT structures and we give two arguments for this fact.
For the first argument, we observe that the claim can be proved directly using Theorem 5.15: the Dolbeault cohomology of these manifolds was computed by Borel [18] and, assuming v ≥ u, it is given by where a p,q is a generator of bidegree (p, q).Hence, for all M u,v , with exception of the three cases known to admit SKT structures, the hypothesis of Theorem 5.15 hold but H 2 (M u,v ) = {0}, hence these manifolds can not be symplectic.
The second argument works for u = 0 and amounts to proving that in this case, the d Hcohomology of M u,v is not isomorphic to the ∂ i∂ω -cohomology, and hence Corollary 5.13 fails.To prove this claim, we start considering the case v ≥ u > 1.If such a manifold had an SKT structure, then [d c ω] = 0 and the d H -cohomology would be isomorphic to the de Rham cohomology.From (5.9), one sees that H 2,1 ∂ (M u,v ) = {0} and hence ∂ω would be the trivial Dolbeault class and the ∂ i∂ω -cohomology would be isomorphic to the usual Dolbeault cohomology.Since the Dolbeault cohomology of the Calabi-Eckman manifolds is not isomorphic to the de Rham cohomology, M u,v can not be SKT.The same argument holds for v > u = 1, except that now H 3 (M v,1 ) = R and there are two possibilities for [H]: it is either zero or not.In the former case, the d H -cohomology is isomorphic to the de Rham cohomology and in the latter it vanishes completely.On the other hand, the Dolbeault cohomology is still described by (5.9) and, as before, H 2,1 ∂ (M ) vanishes so the ∂ i∂ω -cohomology is isomorphic to the ∂-cohomology which is not isomorphic to either the de Rham cohomology or the trivial one.

Hermitian symplectic structures
A particular type of SKT structure for which Corollary 5.13 is particularly relevant are the so called Hermitian symplectic structures [8,23].These consist of a pair (I, ω) of integrable complex structure and symplectic structure such that ω(X, IX) is positive for every nonzero vector X.The difference between these structures and Kähler structures is that here we do not require ω to be of type (1,1).
Theorem 5.17.In a compact Hermitian symplectic manifold the Dolbeault and the de Rham cohomologies are isomorphic as graded vector spaces, i.e., the Frölicher spectral sequence degenerates at the second page.
Proof.Given a Hermitian symplectic manifold, we can decompose ω into its (p, q) components with respect to the complex structure and then one readily obtains that Therefore (I, ω 1,1 ) is an SKT structure and H = id(ω 2,0 − ω 0,2 ) is exact hence the d H -cohomology is isomorphic to the de Rham cohomology.Further, the same identity shows that H 1,2 = −i∂ω 0,2 is ∂-exact hence the ∂ −i∂ω 1,1 -cohomology is isomorphic to the ∂-cohomology.Therefore the result follows from Corollary 5.13.

Hodge theory beyond U (n)
The decomposition of harmonics into their (p, q)-components in a Kähler manifold is a phenomenon that repeats itself for any other special holonomy group and, as such, is a result on Riemannian geometry.This approach is quite different from what we have done so far as, just like in the original Kähler identities, we relied on the underlying complex structures heavily to develop our theory.This section we show that with the appropriate setup, our results can also be extended to any other holonomy groups.Throughout this section we let {e 1 , • • • , e m } ∈ Γ(T M) be an orthonormal frame and {e 1 , • • • , e m } ∈ Γ(T * M ) be its dual frame.
As before, given an oriented Riemannian manifold with closed 3-form, (M, g, H) we consider the metric connection ∇± whose torsion is ∓H and let ∇ denote the Levi-Civita connection.Using the orthonormal frame, we can write explicit expressions for ∇ ± .Indeed, if we define h ijk = H(e i , e j , e k ), then, using Einstein summation convention and omitting the symbol for the wedge product, H is given by and we have If the holonomy of ∇ + is the Lie group G + , then using the isomorphism T M ∼ = V + , we realize its Lie algebra, g + , as a sub Lie algebra of so(V + ) = ∧ 2 V + .Mutatis mutandis, the same holds for ∇ − and we get g − ⊂ ∧ 2 V − .Next we notice that g + ⊕ g − ⊂ ∧ 2 T M = spin(T M ) and hence the elements of g + ⊕g − act on forms, thought of as spinors.Further, since V + is orthogonal to V − and g ± ⊂ ∧ 2 V ± , the Lie algebra action of g + and g − on forms commute and we get an action of g + ⊕g − on forms as a direct sum of the individual actions of the Lie algebras.Now we are in condition to state the main theorem of this section, which extends Theorem 5.7 to general holonomy groups.Theorem 6.1.Let (M, g, H) be a compact Riemannian manifold endowed with a closed 3-form.If ∇ ± , the metric connections with torsion ∓H, have holonomy in G ± , then the d H -cohomology of M splits according to the decomposition of forms into irreducible representations of the action of g + ⊕ g − ⊂ ∧ 2 V + ⊕ ∧ 2 V − on forms.Lemma 6.2.The connections ∇ ± preserve the irreducible representations of g ± , respectively.
Proof.Indeed, if we denote by g + ⊂ spin(T M) ∼ = ∧ 2 T * M the bundle of endomorphisms of T M defined by the connection ∇ + , the condition that ∇ + has reduced holonomy implies that this bundle is preserved by parallel transport.Now g + is simply the image of g + by the parallel isomorphism Id + g : T * M −→ V + , hence the bundle g + ⊂ ∧ 2 V + is also parallel and its irreducible representations are preserved by the connection.

Next we define
and To prove the theorem we need to extend to the torsion case the formulas relating the Levi-Civita connection with the exterior derivative and its adjoint: To avoid considering separate cases according to whether the dimension of the manifold is even or odd, we let Lemma 6.3.With the definitions above Proof.Since the statements are similar, we will only prove the first identity.The left hand side of (6.3) is Next, we use relation (6.1) to re-write the right hand side as H − preserves the irreducible representations of Clif(V + ) ⊕ g − and hence Δ d H preserves the intersections of these representations, which are just the irreducible representations of g + ⊕ g − .

Integrability
A simple consequence of (6.2) is that if W ⊂ ∧ k T * M is a representation of the holonomy group of the Levi-Civita connection, then the exterior derivative restricted to sections of W can only land in representations present in T * M ∧ W .This is the Riemannian version of the claim that in a complex manifold d : Ω p,q (M ) −→ Ω p+1,q (M ) ⊕Ω p,q+1 (M ).Lemma 6.3 has similar implications.Proposition 6.4.Let (M, g, H) be a compact oriented Riemannian manifold with a closed 3-form and let g ± ⊂ ∧ 2 V ± be the Lie algebras of the holonomy groups of the connections ∇ ± .Let W be a representation of g + ⊕ g − , then d H sends sections of W into sections of representations that appear in (Clif Proof.It follows from Lemmata 6.2 and 6.3 that It follows from the material in Sections 3 and 4 that the difference between an SKT or generalized Kähler structure and parallel (almost) Hermitian or (almost) bi-Hermitian structure, respectively, is that in the former two cases if W is a representation of g + ⊕ g − then d H : Γ(W ) −→ Γ(T C M •W ), while in the latter two cases Proposition 6.4 is the best one can say.This suggests that in general there is a subclass of the space of manifolds (M, g, H) with reduced holonomy which may be of further interest: Definition 6.5.We say that (∇ + , ∇ − ) induces an integrable G + × G − structure if the holonomy of ∇ ± is G ± and for every representation W ⊂ ∧ • T * C M of g + ⊕ g − .

Instantons over complex surfaces
Next we use the techniques developed in this section to provide an alternative description of 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 [2] and the SKT reduction theorem as presented in [5].The argument presented here follows closely the one from [3], 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 [13], there is a representative g of the conformal class which makes (M, g, I) into an SKT manifold, i.e., the corresponding Hermitian form ω satisfies dd c ω = 0. We let H = d c ω and consider T M endowed with the H-Courant bracket, so that the metric G = 0 g −1 g 0 and the complex structure on V + induced by I via the isomorphism π : V + −→ T M are a positive SKT structure on T M .Given a bundle E over M with a compact Lie group G as structure group and Lie algebra g we let A be the space of all g-connections on E endowed with the trivial 3form, so that A is an affine space isomorphic to space of 1-forms on M with values in the adjoint bundle g E , Ω 1 (M ; g E ).Hence at any connection A we have T A A = Ω 1 (M ; g E ) and, letting κ be a bi-invariant metric on G, we can use κ to identify T * A A = Ω 3 (M ; g E ).Indeed, for X ∈ Ω 1 (M ; g E ) and ξ ∈ Ω 3 (M ; g E ) we define the natural pairing as ξ(X) = 2 M κ(X, ξ).
Then for X, Y ∈ Ω 1 (M ; g E ) and ξ, η ∈ Ω 3 (M ; g E ) we have where κ(•, •) Ch indicates that one uses κ to pair elements in g E and the Chevalley pairing on forms to obtain a top degree form.
We denote by G the gauge group of E and by g = Ω 0 (M ; g E ) its Lie algebra.The infinitesimal generator corresponding to γ ∈ Ω 0 (M ; g E ) at a point A ∈ A is just the vector d A γ ∈ Ω 1 (M ; g E ) and we can extend this action to form a lifted action as in [5]: as long as there are no infinitesimal symmetries, i.e., as long as d A : Ω 0 (M ; g E ) −→ Ω 1 (M ; g E ) has trivial kernel.Therefore, from this point onwards we only consider connections for which d A : Ω 0 (M ; g E ) −→ Ω 1 (M ; g E ) has trivial kernel.If E is a simple SU (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 values on the G -module h * = Ω 2 + (M ; g E ), the space of self-dual 2-forms: where F A is the curvature of the connection A and (•) + indicates projection onto the space of self dual forms, Ω • + (M ; g E ), which, for 2-forms in 4 dimensions, agrees with the usual self dual 2-forms.If we let a be the sum g ⊕ h, a becomes a Courant algebra if we endow it with the hemisemidirect product: (γ 1 , λ 1 ), (γ 2 , λ 2 ) = ([γ 1 , γ 2 ], γ 1 • λ 2 ).
Then the maps Ψ and μ together give rise to an extended action given infinitesimally by map of Courant algebras: We notice that Ω 0 (M ; g E ) + Ω 2 + (M ; g E ) is isomorphic to Ω ev + (M ; g E ) via the map so we can use a = Ω ev + (M ; g E ) and then the extended action is given simply by Following the reduction procedure, the reduced manifold, M, is obtained by taking the quotient of μ −1 (0) by the action of the gauge group.In this case, μ −1 (0) consists 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 + = W 2 ∩ Ω od (M ; g E ) and V 0,1 + = W −2 ∩ Ω od (M ; g E ).Further, it is immediate that this SKT structure is invariant by the action of the gauge group.Now, we are in position to use the SKT reduction theorem: Theorem 7.1 (SKT Reduction Theorem [5]).Let Ψ : a −→ Γ(T A) be an extended action with 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 moment map, then M red = P/G is smooth and the SKT structure on A reduces to an SKT structure on M red if and only if K ⊥ ∩ V + | P is invariant under I.
As we observed earlier, at a point A ∈ A, K ⊥ corresponds to the d H A+ -closed odd forms and hence K ⊥ ∩ V + corresponds to the d H A+ -closed SD odd forms, that is the SD, d H Aharmonic odd forms and hence according to the theorem to prove that the moduli space of instantons inherits an SKT from M , we must prove that the space of SD, d H A -harmonic odd forms is invariant under I.To achieve this, one must develop Hodge theory for forms with coefficients.In this case, the condition that the connection is anti self-dual, allows us to achieve the result.
Indeed, similarly to the flat case, we can split d H A as a sum of three operators: Indeed, locally d H A = d H + A, for some A ∈ Ω 1 (M ; g E ) and hence the splitting of d H together with the splitting of A into its V 1,0 + , V 0,1 + and V − components gives the desired decomposition of d Hence the Laplacian leaves the spaces W 2 ∩ Ω od and W −2 ∩ Ω od invariant.Therefore we can decompose the space of harmonic SD odd forms into two spaces H ±2 = ker( H ) ∩ W ±2 and I acts as multiplication by i on W 2 and by −i on W −2 , so either way it preserves the intersection ker( H ) ∩ W ±2 and hence it preserves the space of SD harmonic odd forms.According to Theorem 7.1, this means that the SKT structure from A reduces to an SKT structure on M so we have re-obtained the following result, originally due to Lübke and Teleman [21]: an orientation.Then we let {e 1 , e 2 , • • • , e m } be a positive orthonormal basis of V + and let = −e m •• • • e 2 •e 1 ∈ Clif(V ).Then ϕ := • ϕ, where • denotes Clifford action.

Fig. 3 .
Fig. 3. Representation of the nontrivial components of d H for a generalized Hermitian structure.

Fig. 4 .
Fig. 4. Representation of the nontrivial components of d H for a generalized Kähler structure.

Fig. 5 .
Fig. 5. Representation of the nontrivial components of d H for an almost generalized Hermitian structure extending a parallel positive Hermitian structure.

Fig. 6 .
Fig. 6.Representation of the nontrivial components of d H for a parallel bi-Hermitian structure.

Fig. 7 .
Fig. 7. Representation of the nontrivial components of d H for a positive SKT structure.

Fig. 8 .
Fig. 8. Decomposition of d H for a generalized complex extension of a positive SKT structure.

Theorem 5 . 7 .
Let (M, H) be a compact manifold with 3-form, let (G, I + , I − ) be a generalized metric with a pair of almost Hermitian structures andJ 1 = I + + I − .Then 1.If I + (resp.I − ) is a parallel positive (resp.negative) Hermitian structure, the d H -Laplacian preserves the spaces W k,l + (resp.W k,l − ) and hence the d H -cohomology of M inherits a corresponding Z × Z 2 -grading; 2. If (G, J 1 ) is a parallel bi-Hermitian structure, the d H -Laplacian preserves the spaces U p,q and hence the d H -cohomology of M inherits a corresponding Z 2 -grading; H A .Just as in Section 5.2, integration by parts gives that (δN + ) * = −δ N Now, let ϕ ∈ (W 2 ⊕ W −2 ) ∩ Ω od (M ; g E ).Then the d H A -Laplacian computed on ϕ is given byH ϕ = (d H * A d H A+ + d H A d Corollary 4.4.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 Kähler structure if and only if Proposition 4.3.Let G be a generalized metric on a manifold with 3-form (M, H) and let 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 N 2 , N 3 and N + vanish and (G, I − ) is a negative SKT structure if and only if N 1 , N 4 and N − vanish.
1,0 + , one can readily check that ∂ and δ+ have the same symbol and hence the same index.Of course complex conjugation swaps ∂ and ∂, allowing us to translate Theorem 5.6 to Dolbeault cohomology terms: 1  2h ijk e i e j e k −1  2h ijk e i e j e k + H + .Due to (6.2), the first two terms are d − d * .Expanding H + we getε i + ∇ + e i + H + = d − d * − 1 2 h ijk e i e j e k −1 2 h ijk e i e j e k + 1 3! h ijk e i e j e k + 1 2! h ijk e i e j e k + 1 2! h ijk e i e j e k + 1 3! h ijk e i e j e k , which equals(6.4).Proof of Theorem 6.1.Since ∇ + preserves the irreducible representations of g + , Lemma 6.3 implies that / D H + preserves the irreducible representations of g + ⊕ Clif(V − ) and hence so does the d H -Laplacian. Similarly, / D