Singular Reduction of Generalized Complex Manifolds

In this paper, we develop results in the direction of an analogue of Sjamaar and Lerman's singular reduction of Hamiltonian symplectic manifolds in the context of reduction of Hamiltonian generalized complex manifolds (in the sense of Lin and Tolman). Specifically, we prove that if a compact Lie group acts on a generalized complex manifold in a Hamiltonian fashion, then the partition of the global quotient by orbit types induces a partition of the Lin-Tolman quotient into generalized complex manifolds. This result holds also for reduction of Hamiltonian generalized K\"ahler manifolds.


Introduction
Generalized complex geometry was introduced by Hitchin in [11], and further developed by his student Gualtieri in his doctoral thesis [9]. It serves as a common ground in which symplectic, Poisson, and complex geometry can meet. For this reason, there has been much effort to import ideas and techniques from these other fields into the generalized complex setting. In particular, many constructions and results from equivariant symplectic geometry have found useful analogues here. One example is that of Hamiltonian group actions and moment maps, developed in [14]. (Similar constructions were developed and examined by other groups, such as [4,20] and [12], but this paper expands specifically on the work of Lin and Tolman. ) Lin and Tolman's construction generalizes the usual symplectic definition, and they go on to prove that one can reduce a generalized complex manifold by its Hamiltonian symmetries, in perfect parallel to Marsden-Weinstein symplectic reduction [15] (sometimes also credited to Meyer [16]). Just as in the symplectic case, in order to ensure that the generalized reduced space is a manifold, one must make an assumption regarding freeness of the group action.
In [19], Lerman and Sjamaar proved that if the symplectic reduced space is not a manifold, then the orbit type stratification of the original symplectic manifold induces the structure of a stratified space (see Definition 1.7 of that paper) on the reduced space, each stratum of which is naturally a symplectic manifold. The main result of this paper, Theorem 5.3, is a first step in the direction of an analagous result for the case of a singular generalized complex reduced space. It states that the singular generalized complex reduced space can be partitioned into disjoint generalized complex manifolds. It is not yet known whether the reduced space in this situation is actually a stratified space. A similar, although distinct, situation was studied in [13], in which the authors considered the singular reduction of Dirac manifolds. They analyzed the global quotient of a Dirac manifold by a proper group action as a differential space, as in [6], and obtained conditions that guarantee the Dirac structure will descend to the quotient space.
An interesting difference between the symplectic and generalized complex situations is that the generic result of symplectic reduction is a space with at worst orbifold singularities, whereas for the reduction of a twisted generalized complex manifold, the generic result may be a space with worse-than-orbifold singularities. See Remark 5.1 below. Section 2 is a rapid introduction to some essential notions from generalized complex geometry. Section 3 reviews some important facts about equivariant cohomology and the orbit type stratification of G-spaces. Section 4 consists of a summary of Hamiltonian actions and reduction in generalized complex geometry. Finally, Section 5 contains the full statement and proof of our main theorem.
An earlier version of this work appeared in the author's doctoral thesis [8], where many definitions and calculations are explained in great detail.
Throughout, we use the abbreviations "GC" for "generalized complex" and "GK" for "generalized Kähler". Also, we typically use the same notation and nomenclature to refer to both a map and its complex linear extension. Finally, we make use of the musical notation for the map between a vector space and its dual induced by a bilinear form. If B : V × V → R is a bilinear form on a real vector space V , then we will denote by B ♭ : V → V * the map v → ι v B := B(v, ·), where ι v denotes the interior product by v. If B is non-degenerate, then B ♭ is invertible and we denote its inverse by B ♯ := B ♭ −1 . We also use the musical notation for vector bundles, sections of their second symmetric powers, and the associated bundle maps.

Generalized complex geometry
We begin by giving several standard definitions and results from generalized complex geometry, which can be found in [9] or [3].
For any smooth manifold M , the Pontryagin bundle, or generalized tangent bundle, of M is TM := T M ⊕ T * M . This vector bundle carries a natural non-degenerate symmetric metric ·, · of signature (n, n), defined by for all x ∈ M and u + α, v + β ∈ T x M . We will use the same notation for the complex bilinear extension of this metric to the complexification T C M := TM ⊗ R C. These metrics will henceforth be referred to as the standard metrics on TM and T C M . Let J 1 and J 2 be commuting almost GC structures on M . Then G := −J 1 • J 2 is an orthogonal and involutive bundle map TM → TM , and there is an associated bilinear form defined by for all X , Y ∈ TM in a common fiber. We call G positive definite if its associated bilinear form is positive definite, i.e. if G(X ), X > 0 for all nonzero X ∈ TM . An almost GK structure on M is a pair of commuting almost GC structures (J 1 , J 2 ) on M such that G := −J 1 • J 2 is positive definite.  for an almost GC structure given equivalently by a map J or a Dirac structure E, and it preserves types. It transforms almost GK structures (J 1 , J 2 ) by transforming J 1 and J 2 individually.
The Lie bracket defines a skew-symmetric bilinear bracket on sections of the tangent bundle T M . This can be extended to a skew-symmetric bilinear bracket on sections of the Pontryagin bundle TM , called the Courant bracket, defined by for all X + α, Y + β ∈ Γ(TM ), where Γ(TM ) denotes the space of smooth sections of TM → M .
Here the bracket on the right-hand side is the usual Lie bracket of vector fields, and L denotes Lie differentiation. For each closed differential three-form H ∈ Ω 3 cl (M ), there is also the H-twisted Courant bracket, defined by . Both the Courant and the H-twisted Courant brackets extend complex linearly to brackets on smooth sections of the complexified Pontryagin bundle T C M , which will be denoted the same way. Let E ⊂ T C M be an almost GC structure on M . This is a GC structure if E is Courant involutive, in which case (M, E) is a GC manifold . If H ∈ Ω 3 cl (M ) and E is H-twisted Courant involutive, then E is an H-twisted GC structure, and (M, E, H) is a twisted GC manifold .
Let (J 1 , J 2 ) be an almost GK structure on M . This is a GK structure if both J 1 and J 2 are Courant involutive, in which case (M, J 1 , J 2 ) is a GK manifold . If H ∈ Ω 3 cl (M ) and J 1 and J 2 are H-twisted Courant involutive, then this is an H-twisted GK structure, and (M, J 1 , J 2 , H) is a twisted GK manifold .  1. Let (M, ω) be an almost symplectic manifold , meaning that ω ∈ Ω 2 (M ) is a nondegenerate form on M , but not necessarily closed. This defines an almost GC structure J ω on M by of type 0 at every point. It has associated Dirac structure defined by for each x ∈ M . As discussed in Section 3 of [9], it is a GC structure if and only if dω = 0, i.e. if and only if ω is a symplectic structure on M .
2. Let (M, I) be an almost complex manifold , meaning that I 2 = −id T M but I is not necessarily integrable. This defines an almost GC structure J I on M by of type n at every point. It has associated Dirac structure defined by where T 1,0 M, T 0,1 M ⊂ T C M denote the ±i-eigenbundles of I. As discussed in Section 3 of [9], it is a GC structure if and only if I is integrable, i.e. if and only if I is a complex structure on M .
3. Let M be a Kähler manifold with Kähler form ω ∈ Ω 2 (M ), complex structure I : T M → T M , and associated Riemannian metric g. Then J ω and J I commute and Example 2.10. Let (M 1 , J 1 ) and (M 2 , J 2 ) be almost GC manifolds. Then the direct sum J of J 1 and J 2 is a map J := (J 1 , J 2 ) : TM 1 ⊕TM 2 → TM 1 ⊕TM 2 , which under the identification TM 1 ⊕ TM 2 ∼ = T(M 1 × M 2 ) yields an almost GC structure on M 1 × M 2 . We will call this the direct sum of the almost GC structures on M 1 and M 2 . It is not hard to see that (J 1 , J 2 ) is a GC structure on M 1 × M 2 if and only if J i is a GC structure on M i for i = 1, 2.
Let H 1 ∈ Ω 3 cl (M 1 ) and H 2 ∈ Ω 3 cl (M 2 ), let π i : M 1 × M 2 → M i be the natural projection for i = 1, 2, and set H := π * 1 H 1 + π * 2 H 2 . By the naturality of the exterior derivative, we know H is a closed three-form on M 1 × M 2 . Furthermore, it is not hard to see that ( There is a completely analogous product construction for almost GK and GK manifolds as well. Let (M, E, H) be a twisted GC manifold. Suppose S is a submanifold of M given by the embedding j : S ֒→ M . Although j induces a natural embedding j * : T S ֒→ T M of tangent bundles, because of the contravariance of cotangent bundles there is in general no obvious embedding TS ֒→ TM of the Pontryagin bundles. The following definition comes from [3].
For each x ∈ S, define Then E S is a constant-rank complex linear distribution of T C S, but is not in general a smooth subbundle, nor will it generally satisfy E S ∩ E S = 0.  The following is an extension of Proposition 5.12 of [3] to the twisted case. As with Proposition 2.11, the original proof still holds with only minor alterations.
It is straightforward to show that this implies the following.
Definition 2.17. Let M be a manifold, and let G be a Lie group acting smoothly on M . This lifts to an action of G on TM by bundle automorphisms, given by where g * is the pushforward of tangent vectors by the map g : M → M and (g −1 ) * is the pullback of tangent covectors by the map g −1 : M → M .
Let J be an H-twisted GC structure on M . We say that the G-action on (M, J , H) is canonical if the following hold.
2. The action of G on TM commutes with J , i.e. the diagram It is easy to check that a smooth group action on a manifold commutes with an almost GC structure J : TM → TM if and only if the complex linear extension of the action preserves the corresponding complex Dirac structure. Proof . Let (M G ) ′ be a component of M G . First, recall that for each x ∈ (M G ) ′ the derivative of the action of G at x defines a linear action of G on T x M , and that T x (M G ) ′ = (T x M ) G . Let dg be a bi-invariant Haar measure on G, adjusted so that dg(G) = 1. Define a bundle map π : That π is a bundle map and N is a vector bundle follow from the naturality of the technique of averaging by integration. Note also that T and hence a proof of this claim completes the proof of this proposition.
Let λ ∈ Ann(W ), let g ∈ G, and let u ∈ V . Decompose u as

Background information on G-spaces
In this section we give some brief definitions and results about compact group actions on manifolds which will be required in later sections. The standard reference for the material on equivariant cohomology is [10]. The material on orbit spaces and their stratification by orbit types can be found in [7, Chapter 2] and [18, Chapter 2].

Equivariant cohomology
Let M be a manifold and G be a compact Lie group acting smoothly on M . Consider the space Ω k (M ) ⊗ S i (g * ), where S i denotes the degree i elements of the symmetric algebra. This is a Gspace with action defined by linear extension of the rule g · (α ⊗ p) := (g −1 ) * α ⊗ (p • Ad g −1 ) for g ∈ G, α ∈ Ω ⋆ (M ), p ∈ S(g * ). We can identify Ω k (M ) ⊗ S i (g * ) with the space of degree i polynomial maps g → Ω k (M ) via α ⊗ p : ξ → p(ξ) · α for ξ ∈ g. An element of Ω k (M )⊗S i (g * ) is G-invariant if and only if its corresponding polynomial map is G-equivariant with respect to the adjoint action of G on g and the action of G on Ω k (M ) given by g · α := (g −1 ) * α for g ∈ G, α ∈ Ω k (M ). i=0 The differential d G : Ω n G → Ω n+1 G is defined, viewing equivariant forms as maps g → Ω ⋆ (M ), by The    2. θ(ξ M ) ≡ ξ for all ξ ∈ g.

Orbit type stratification
Let G be a group. For each subgroup H of G, we will denote by (H) the set of subgroups of G that are conjugate to H. Suppose G is a compact Lie group and M is a manifold on which G acts smoothly. Note that the conjugacy relation among subgroups of G preserves closedness, and hence also preserves the property of being a Lie subgroup. Some important properties of the sets we have defined above are collected in the following proposition. Their proofs can be found in the references cited at the beginning of this section. In general, the orbit space M/G can be a very singular space. It will be a Hausdorff and second-countable topological space, but will rarely inherit a manifold, or even an orbifold, structure from M . However, because M is the disjoint union of its orbit type submanifolds, we can also partition the orbit space: where the disjoint union is taken over all the distinct orbit type submanifolds of M . Since each component of M (H) /G is a manifold, we know that, after refining the partition to components, (3.1) is a partition of M/G into manifolds. It is called the orbit type partition of M/G.

Hamiltonian actions on generalized complex manifolds
In [14], the authors proposed the following definition of Hamiltonian actions on GC manifolds.
Here µ ξ is as defined above, and α ξ ∈ Ω 1 (M ) is the differential one-form on M defined by   (a) Note that a moment one-form α ∈ Ω 1 (M, g * ) is an equivariant differential form of degree 3.
(b) Because E is an isotropic subbundle, the condition that ξ M + α ξ − idµ ξ ∈ E implies that ξ M + α ξ − idµ ξ , ξ M + α ξ − idµ ξ = 0, and hence that ι ξ M α ξ = ι ξ M dµ ξ = 0.  Recall that this means the G-action is symplectic, the map Φ is G-equivariant, and for all ξ ∈ g we have dΦ ξ = ι ξ M . Let J ω be the GC structure on M induced by ω. As discussed in Example 3.8 of [14], the action of G on (M, J ω ) is generalized Hamiltonian, and Φ is a generalized moment map. Proof . First we will prove that the action of G on S preserves E S . Let x ∈ S and (X, λ) ∈ (T C,x S ⊕ T * C,x M ) ∩ E x , which means that (X, λ| S ) ∈ E S,x . Then for any g ∈ G we have g · (X + j * λ) = g * (X) + (g −1 ) * λ| S .
Because S is G-stable, the inclusion j : S ֒→ M is G-equivariant, i.e. the G-action commutes with j. Hence j * : T S ֒→ T M is G-equivariant, so g * (X) ∈ T g·x S. Also Since E is G-stable, we have g · (X + λ) = g * (X) + (g −1 ) * λ ∈ E g·x . Therefore g · (X, j * λ) = g * (X), j * (g −1 ) * λ ∈ E S,g·x . Thus E S is G-stable. Now suppose that (S, E S , j * H) is a GC submanifold of (M, E, H), meaning that E S is a vector bundle, that E S ∩ E S = 0, and that E S is j * H-twisted Courant involutive. Since j is Gequivariant, for all ξ ∈ g we have ξ M | S = ξ S , (j * µ) ξ = j * (µ ξ ), and (j * α) ξ = j * (α ξ ). Furthermore, by the naturality of the exterior derivative we have Again using the G-equivariance of j, for all x ∈ S we have Thus the action of G on (S, E S , j * H) is twisted Hamiltonian with moment map µ| S and moment one-form α| S .
The above result holds also for the untwisted case, of course, by putting H = 0 and α = 0.
The following three results are exactly what makes reduction of generalized Hamiltonian manifolds possible.      [14], in the context of the hypotheses of Theorem 4.6, if the GC structure and moment map come from a symplectic structure and moment map, then the GC structure on the quotient is exactly the one induced by the Marsden-Weinstein ssymplectic structure on the quotient.

M ) is closed and basic and so descends to a closed form
The following result will be useful to us later. Its proof follows trivially from the definitions of generalized and twisted generalized Hamiltonian actions.
Embedding G diagonally in G × G, we obtain a Hamiltonian action of G on M 1 × M 2 . The projection g * ⊕ g * ։ g * induced by this embedding is given by addition: (λ 1 , λ 2 ) → λ 1 + λ 2 , so a moment map and moment one-form for the G-action on M 1 × M 2 is given by and respectively.
Perhaps the most important instance of the construction of Example 4.12 is if we start with an arbitrary twisted generalized Hamiltonian G-manifold, (M, J , H, µ, α), and let the second GC manifold be a coadjoint orbit O a in g * , where a ∈ g * is some fixed element. Let ω a be the canonical symplectic structure on O a . The action of G on O a is Hamiltonian in the symplectic sense, with moment map given by the inclusion O a ֒→ g * . Using the symplectic structure −ω a instead, the action is still Hamiltonian, but now the moment map is given by the negative inclusion O a → g * , λ → −λ.
As described in Examples 2.9 and 4.4, the symplectic structure −ω a induces a GC structure J a on O a , and the G-action on O a is generalized Hamiltonian with the same moment map. Viewing (O a , J a ) as a twisted GC manifold where the twisting is by the zero three-form, the G-action is twisted generalized Hamiltonian with a constantly vanishing moment one-form. Then the diagonal G-action on M × O a is twisted generalized Hamiltonian with moment map and moment one-form The reason this construction is important is that it is the basis of the shifting trick . If one wishes to reduce M by G at level a ∈ g * , one can instead consider the reduction of M × O a by G at level 0, because as topological spaces. To see this, observe that µ −1 (O a ) and (µ ′ ) −1 (0) are G-equivariantly homeomorphic via the maps Suppose now M is a symplectic manifold, the G-action is Hamiltonian, and µ is a moment map. In this case the quotient space M a := µ −1 (O a )/G is called the symplectic reduction, or Marsden-Weinstein quotient, of M at level a. The symplectic moment map condition is that dµ ξ = ι ξ M ω for all ξ ∈ g. If G acts freely on µ −1 (O a ), then each ξ M is nonzero on µ −1 (O a ), which by the non-degeneracy of ω implies that a is a regular value of µ. Therefore µ −1 (O a ) ⊂ M is a submanifold, so M a is a manifold. In this case, Marsden and Weinstein proved that M a inherits a natural symplectic structure. Theorems 4.6 and 4.8, proved in [14], are analogues of this result.
In the event that the symplectic quotient is singular, one can consider the individual parts of the partitioned quotient. In [19], Lerman and Sjamaar proved that each component of (M a ) (H) := µ −1 (O a ) ∩ M (H) /G inherits a natural symplectic structure. The main results of this paper are analogues of this in the generalized complex case.
Remark 5.1. By the symplectic moment map condition, dµ ξ = ι ξ M ω, if a ∈ g * is a regular value of µ, then each vector field ξ M is nowhere zero on µ −1 (a). This means that the action of G on µ −1 (a) is at least locally free, which means that the quotient M a is at worst an orbifold, to which Marsden and Weinstein were able to associate a symplectic structure. By Sard's Theorem, a generic value of µ will be regular, so the generic result of symplectic reduction is a symplectic orbifold.
If (M, J ) is an untwisted GC manifold with moment map µ, then the generalized moment map condition, J (dµ ξ ) = −ξ M , likewise guarantees the equivalence of regular values and local freeness of the action, so M a is at worst an orbifold. However, if (M, J , H) is a twisted GC manifold with moment map µ and moment one-form α, then this equivalence may no longer hold, due to the presence of the moment one-form in the moment condition: Specifically, ξ M could vanish even if J (dµ ξ ) does not. Therefore, it seems that the generic result of GC reduction may be a GC singular space.
Before stating and proving our main theorem, we need the following lemma. In Lemma 5.5 of [1], the authors proved the above result in the case that G is a torus; however, their proof holds just as well in the non-abelian case. It relies on viewing the components of µ as the real parts of a pseudo-holomorphic function and applying a version of the Maximum Principle, a course first taken in [17]. A thorough description of this version of the Maximum Principle can be found in Section 4.4 of [8].   Proof . We begin with the twisted case. First we prove the theorem in the case that a = 0. Let x ∈ M and K = G x . Note that this implies that K is a closed subgroup of G, and is hence compact. Clearly has the same dimension as M lx K . Let M K x be the union of components of M K having nontrivial intersection with M lx K . Since each component of M K is a manifold, it follows that M K x is also. Furthermore, by Proposition 2.19 we know that each connected component of M K is a split submanifold of (M, J ), and hence also a twisted GC submanifold. Therefore so is M K x . Let Z x K be the union of components of M lx K that have nontrivial intersection with µ −1 (0). Since Z x K is open in M lx K , which is open in M K x , as discussed in Remark 2.13 we know that Z x K is a twisted GC submanifold of M K x , and hence also of M . Let j : Z x K ֒→ M be the inclusion, and denote the (j * H)-twisted GC structure of Z x K by J ′ . Let N = N G (K) be the normalizer of K in G. By part (e) of Proposition 3.6, we know M lx K is N -stable. In fact, so is Z x K , as we now show. Note that connected components of manifolds for all [ξ] ∈ (n/k) * ∼ = l * . Hence j * H + α ′ is L-equivariantly closed. Therefore we can apply Lin-Tolman's twisted generalized reduction, Theorem 4.8 above, and obtain a GC structure on the quotient space Recall that this structure is only natural up to B-transform. It follows that each component of Z x K ∩ µ −1 (0) N is a twisted GC manifold. By varying the point x ∈ M K , and thus varying the manifold Z x K , we can conclude that every component M K ∩ µ −1 (0) N is a twisted GC manifold.
By parts (d) and (f) of Proposition 3.6, we know that G · M K = M (K) and that the inclusion M K ֒→ M (K) induces a homeomorphism M K /N ≈ M (K) /G. Together with the fact that µ −1 (0) is G-stable, this first fact implies that G · M K ∩ µ −1 (0) = M (K) ∩ µ −1 (0). Together with the second fact, this implies that and so each component of (M 0 ) (K) inherits a twisted GC structure, natural up to B-transform.
The general case, where the reduction is taken at an arbitrary level a ∈ g * now follows from the shifting trick, as explained following Example 4.12 above. Now we consider the untwisted case. Since an untwisted Hamiltonian GC manifold is simply a twisted Hamiltonian GC manifold with H = 0 and α = 0, the only real difference between parts (a) and (b) of this theorem is that in part (a) we do not assume that M is compact. Note that the only time above where we used the fact that M is compact was when showing that µ ξ and α ξ both vanish on Z x K for all ξ ∈ k, and hence that µ(Z x K ) and α(T Z x K ) lie in Ann g * (k). For this non-compact case, note that since Z x K contains only K-fixed points, we have ξ Z x K = 0 for all ξ ∈ k, so dµ ξ = J ′ (ξ Z x K ) = J ′ (0) = 0 and hence µ ξ is locally constant on Z x K for all ξ ∈ k. Because each component of Z x K has nonempty intersection with µ −1 (0), it follows that µ ξ = 0 for all ξ ∈ k, so µ(Z x K ) ⊂ Ann g * (k). This completes the proof of (a). Proof . Suppose (M, J 1 , J 2 ) is a GK manifold, twisted or untwisted. Because a generalized Hamiltonian action on the GK manifold (M, J 1 , J 2 ) is simply a generalized Hamiltonian action on the GC manifold (M, J 1 ) which also preserves the structure J 2 , it is easy to check that the proof of Theorem 5.3 holds in precisely the same way for our present situation. We will simply note that, for any Lie subgroup K of G, because both J 1 and J 2 are preserved by K, by Proposition 2.19 we know that each component of M K is a split submanifold of M with respect to both GC structures, so it is a GK manifold. Everything else is entirely straightforward to check.