Deformation classes in generalized K\"ahler geometry

We introduce natural deformation classes of generalized K\"ahler structures using the Courant symmetry group. We show that these yield natural extensions of the notions of K\"ahler class and K\"ahler cone to generalized K\"ahler geometry. Lastly we show that the generalized K\"ahler-Ricci flow preserves this generalized K\"ahler cone, and the underlying real Poisson tensor.


Introduction
A rudimentary notion of Kähler geometry is that of the Kähler class: given (M 2n , ω, J) a Kähler manifold, the Kähler form ω is closed, and [ω] ∈ H 1,1 R is the associated Kähler class. Fixing the complex structure J, the space of all Kähler classes defines an open cone in H 1,1 R (the Kähler cone), and the fundamental result of Demailly-Paun [5] gives a characterization of this cone in terms of pairing against complex subvarieties. The space of Kähler metrics within a given Kähler class is an open infinite dimensional cone in C ∞ (M ) using the √ −1∂∂-lemma. Thus the basic structure of the space of Kähler metrics compatible with a fixed complex structure J is fairly well understood. If we instead ask for the space of all Kähler pairs (g, J) on a given smooth manifold M , the question becomes decidedly more delicate. The global structure can be quite wild, with disconnected components of arbitrarily large dimension (cf. [4]).
Understanding the space of generalized Kähler structures on a given manifold M becomes even more delicate. Originally discovered by Gates-Hull-Rocek [6], a generalized Kähler structure on a smooth manifold M is a triple (g, I, J) consisting of a Riemannian metric g compatible with two integrable complex structures I, J further satisfying Later, Gualtieri [8] gave a natural description of this geometry using the language of Hitchin's generalized complex structures [9], in particular in terms of a pair of generalized complex structures (J 1 , J 2 ) satisfying some natural conditions (cf. §2.1). A fundamental question is to understand the degrees of freedom, moduli, and topology of the space of generalized Kähler structures on a given smooth manifold. Whereas in the Kähler setting we can roughly speaking divide the problem of understanding the space of Kähler metrics into the space of possible complex structures and then to consider the space of compatible Kähler metrics, in the generalized Kähler setting such a decomposition is not really possible. Indeed, in many settings, given two complex structures I, J, there is at most one compatible metric which defines a generalized Kähler structure. Nonetheless, many different classes of deformations of generalized Kähler structure have been constructed. Joyce gave the first examples of nontrivial (i.e. non-Kähler) generalized Kähler structures by deforming away from hyperKähler structures (cf. [1]), specifically using an action of diffeomorphisms which are Hamiltonian with respect to an associated holomorphic symplectic structure. Later Hitchin produced nontrivial generalized Kähler structures on del Pezzo surfaces, with a choice of holomorphic Poisson structure playing a key role [10]. Also, Goto [7] has extended the stability result of Kodaira-Spencer to the generalized Kähler setting, with the restriction that one of the generalized complex structures be defined by a pure spinor.
Date: May 5th, 2020. 1 Our main purpose in this work is to describe a class of deformations which generalizes and unifies different notions of "Kähler class" arising in the different flavors of generalized Kähler geometry. It is well-known that two-forms (B-fields) can act on generalized complex structures by conjugation, with the integrability condition being preserved if and only if B is closed. Our deformations exploit a different, and moreover infinitessimal, action of B-fields. In particular, we will say (cf. Definition 3.1) that a one-parameter family of generalized Kähler structures is a canonical deformation if there exists a one parameter family K t ∈ Λ 2 such that for all times t where defined, one has ∂ ∂t Equivalence classes of canonical deformations lead to natural definitions of generalized Kähler class and generalized Kähler cone (cf. §3). These definitions make nonobvious departures from the classical idea of Kähler class and Kähler cone. The first is the use of infinitessimal deformations as opposed to 'large' deformations. Whereas any two metrics in the same Kähler class admit an explicit relationship using the √ −1∂∂-lemma, we can no longer expect such an explicit relationship in general. For instance, as described above Joyce's construction of nontrivial GK structure uses diffeomorphisms which are Hamiltonian with respect to the associated holomorphic symplectic structure, and in general these cannot be described by a single potential function. Instead, as is typical of Hamiltonian diffeomorphisms, we expect to be able to explicitly describe their infinitessimal deformations. Moreover, given that in the Kähler setting, deformations in the Kähler class involve freezing the complex structure and varying the Kähler form, it is natural to imagine that one should deform while fixing either J 1 , J 2 . Nonetheless through careful consideration of natural variational classes of different flavors of generalized Kähler metrics it emerges that varying J 1 and J 2 simultaneously will correctly capture various existing notions of Kähler class in GK geometry.
A fundamental first step in unpacking this definition is to derive the algebraic and differential conditions which are imposed on K to preserve the compatibility and integrability conditions for the pair (J 1 , J 2 ). Through careful computations, it turns out that the answer is pleasingly simple: is a one-parameter family of generalized almost complex structures such that ∂ ∂t for some one parameter family K t ∈ Λ 2 . Then (J t 1 , J t 2 ) is a one-parameter family of generalized Kähler structures if and only if for all t one has where J t is determined via the Gualtieri map, and , denotes the symmetric neutral inner product on T ⊕ T * (cf. §3.1).
In particular, this theorem exhibits that the canonical deformations are, as is true in the Kähler setting, determined infinitessimally by a closed form which is (1, 1) with respect to J. We emphasize here that the condition that dK = 0 does not follow from the known fact that the conjugation action of B-fields on generalized complex structures preserves integrability if and only if dB = 0. For instance, if we consider our infinitessimal action on a single generalized complex structure, the condition to preserve integrability is strictly weaker than dK = 0 (cf. Proposition 2.3). It is only in the context of preserving the integrability conditions of generalized Kähler geometry that one derives dK = 0.
Despite the simplicity of the conditions of Theorem 1.1 and the apparent simplicity of canonical deformations from the point of view of generalized geometry, the deformations induced on the classical bihermitian data (g, I, J) are delicate. Remarkably, these canonical deformations unify all previously known instances of "Kähler class" in generalized geometry, specifically the classical notion of Kähler class, the modified Kähler classes implicit in Apostolov-Gualtieri ([2] Proposition 5, cf. also [6]) in the commuting GK case, as well as Joyce's Hamiltonian deformation construction in the nondegenerate case. We state this for emphasis (cf. §3.6 for notation): Proposition 1.2. The following hold: (1) Given (M 2n , g, J) a Kähler manifold, and u ∈ C ∞ (M ) such that ω + √ −1∂∂u > 0, the one-parameter family arises as a canonical deformation of generalized Kähler structures for 0 ≤ t ≤ 1 defined by (2) Given (M 2n , g, I, J) a generalized Kähler manifold such that [I, J] = 0, and u ∈ C ∞ (M ) arises as a canonical deformation of generalized Kähler structures for 0 ≤ t ≤ 1 defined by (3) Let (M 2n , g, I, J) be a generalized Kähler manifold such that the Poisson structure σ = 1 2 [I, J]g −1 is nondegenerate, with Ω = σ −1 . Given u t ∈ C ∞ (M ) a family of smooth functions, let φ t denote the one-parameter family of Ω-Hamiltonian diffeomorphisms generated by u t . Then, for all t such that − Im π Λ 1,1 I Ω J > 0, the one-parameter family of generalized Kähler structures determined by Ω t = Ω, I t = I, J t = φ * t J defines a canonical deformation of generalized Kähler structures determined by As a final point to contextualize these deformations, we recall that various interesting deformation classes of generalized Kähler structure have been produced using holomorphic Poisson structures. Given a generalized Kähler structure (g, I, J), there is a Poisson tensor σ = 1 2 [I, J]g −1 which is the real part of a holomorphic Poisson tensor with respect to both I and J. By choosing an appropriate deformation of σ, Hitchin [10] produced deformations of Kähler metrics on del Pezzo surfaces to strictly generalized Kähler structures. Also the deformation theory of Goto [7] changes this underlying Poisson tensor. As it turns out our deformations fix σ and I, so occur against a fixed background of a holomorphic Poisson structure.
is a canonical deformation of generalized Kähler structures. Then for all t, As an application, we are able to express the generalized Kähler-Ricci flow in a simple way using canonical deformations. The equation is an extension of Kähler-Ricci flow to the setting of generalized Kähler geometry, introduced by the second author and Tian [13]. Recently this flow has been used to study the global topology of the (nonlinear) space of generalized Kähler structures in certain settings [3]. To describe this flow, fix (g, I, J) a generalized Kähler structure. Associated to the Hermitian structure (g, I) is the Bismut connection where H = d c I ω I , and D denotes the Levi-Civita connection. This is a Hermitian connection, and if Ω I denotes its curvature, we obtain a representative of the first Chern class via contraction, called the Bismut-Ricci tensor: 2 tr Ω I I. From the Bianchi identity we know that dρ I = 0, but it is not in general true that ρ I ∈ Λ 1,1 I , and we will let ρ 1,1 I denote its (1, 1) projection. Furthermore, associated to (g, I) we obtain the I-Lee form, defined by Similarly we obtain the Lee form θ J associated to (g, J). With this background in place, we can describe the generalized Kähler-Ricci flow in the I-fixed gauge simply by The evolution of the complex structure J is derived in [13], arising from delicate gauge manipulations and curvature identities. On the other hand it has been shown in several special cases (cf. §4 below) that the generalized Kähler-Ricci is driven entirely by ρ I . Using our description of canonical deformations, and a further subtle curvature identity for generalized Kähler manifolds (Proposition 4.2), we confirm that this is true in full generality, and give a very simple description of generalized Kähler-Ricci flow in terms of the associated generalized complex structures.
In other words, the generalized Kähler-Ricci flow is the canonical deformation driven by the I-Bismut-Ricci tensor.
Immediately following from Theorem 1.4 and Corollary 1.3 is that fact that generalized Kähler-Ricci flow preserves the underlying real Poisson tensor σ, and moreover preserves the generalized Kähler cone associated to the initial data. Corollary 1.5. Let (M 2n , g t , I, (J) t ) be a solution of generalized Kähler-Ricci flow in the I-fixed gauge. The associated one-parameter families of generalized complex structures (J 1 , J 2 ) lies in the generalized Kähler cone associated to the initial data. In particular, for all t such that the flow is defined, in other words, the real Poisson tensor σ is fixed along the flow.
2. Formal deformations of generalized complex structure 2.1. Background. Given M a smooth manifold, the generalized tangent bundle is given by T ⊕T * . This bundle comes equipped with a family of natural brackets determined by a closed three-form H. In particular, given H ∈ Λ 3 T * , dH = 0, define the twisted Courant bracket [, ] for sections of A generalized complex structure J is then an almost complex structure on T ⊕ T * , whose √ −1eigenbundle, denoted by L, is integrable with respect to the the twisted Courant bracket. This condition is naturally captured by a corresponding version of the Nijenhuis tensor, where for a given almost complex structure we associate the natural projection maps π 0,1 , π 1,0 and then for x, y ∈ T ⊕ T * we have Direct computations show that this is tensorial, and vanishes if and only if the associated almost generalized complex structure is integrable. See [8] for further discussion.

2.2.
Variations of generalized complex structure. To begin we define an action of B-fields on generalized complex structures.
Definition 2.1. Given a smooth manifold M and K ∈ Λ 2 , define For a given J, we intend to use Φ K (J) as a tangent vector to a one-parameter variation of J through generalized complex structures. We first note that variations of this kind will indeed preserve the space of generalized almost complex structures.
Collecting these computations gives the result.
Corollary 2.4. Let J t be a one-parameter family of generalized almost complex structures such that J 0 is integrable and for all t one has where furthermore for all x, y ∈ T ⊕ T * one has π t 0,1 dK t π T π t 1,0 ( y), π T π t 1,0 ( x), · = 0. (2.5) Then J t are integrable for each t.
Proof. Choosing any Hermitian metric on (T ⊕ T * ) ⊗ C, using Proposition 2.3 and the hypothesis (2.5) one directly derives for all t ∂ ∂t Since N J 0 = 0 the result follows from Gronwall's inequality.

Variations of generalized Kähler structure
Having defined certain variations of generalized complex structure, we now extend this to defining variations of generalized Kähler structure. A naive guess would be that we should simply take a variation of one of the underlying generalized complex structures and seek the further integrability conditions. However, for reasons to be illuminated by the examples below, it is much more natural to vary both generalized complex structures by a single B-field as described in §2.2.

Background.
A generalized Kähler structure is a pair of commuting generalized complex structures J 1 , J 2 such that G = −J 1 J 2 defines a generalized metric, i.e. G·, · is a positive definite inner product on T ⊕ T * , where , denotes the symmetric neutral inner product on T ⊕ T * , i.e. X + ξ, Y + η = 1 2 (ξ(Y ) + η(X)) . A fundamental theorem of Gualtieri ([8] Chapter 6) says that a generalized Kähler structure (J 1 , J 2 ) as defined here corresponds to a bihermitian structure (g, I, J, b), with Kähler forms ω I , ω J , as described in the introduction. The explicit relationship is given by We recall that a generalized Kähler structure induces a fourfold decomposition of the complexified generalized tangent bundle. Specifically, letting L i and L i denote the ± √ −1-eigenbundles of J i respectively, we have the following decomposition: Given J 1 , J 2 another generalized Kähler structure, we define an equivalence relation where if and only if there exists a canonical family (J t 1 , J t 2 ), t ∈ [0, 1], such that (J 0 1 , J 0 2 ) = (J 1 , J 2 ), (J 1 1 , J 1 2 ) = ( J 1 , J 2 ). Furthermore, the generalized Kähler cone associated to (J 1 ,

Compatibility Condition.
We first address the condition required for a canonical deformation to preserve the algebraic compatibility condition of generalized Kähler structures. We first prove a formal lemma reducing this to an algebraic condition on K, then analyze this explicitly using the Gualtieri map.
Lemma 3.2. Let J t 1 , J t 2 be one-parameter families of generalized almost complex structures, with Then for all t.
Since [J 0 1 , J 0 2 ] = 0, the result follows. Here we reformulate the compatibility condition of Lemma 3.2 by expanding the necessary equation in terms of the Gualtieri map and analyzing the result, which simplifies dramatically.
if and only if K ∈ Λ 1,1 J .
Using this notation and the fact that e K and e b commute, it follows easily that Hence As a first step, we record the simplified forms of Φ K (Υ 1/2 ) obtained through a direct computation: .
Further tedious computation yields .
By comparing each entry of the matrices above, we see equality holds if and only if KJ = −J * K, as required.
3.4. Integrability Condition. We next address the integrability condition. Since our deformations should preserve integrability of each generalized complex structure J i , Proposition 2.3 yields two partial integrability conditions which K must satisfy. We again emphasize that neither of these conditions alone will force dK = 0, while somewhat surprisingly the combination of the two conditions does.
Proposition 3.4. Given M a smooth manifold and (J 1 , J 2 ) a generalized Kähler structure, for K ∈ Λ 2 one has Proof. The sufficiency of dK = 0 is obvious, we prove it is necessary. Note that for a pure covector Fix vectors X, Y ∈ T 1,0 I , and then choose lifts X + , Y + to C + , the +1-eigenspace of G. Using the representation of J i with respect to the ±1-eigenspace decomposition induced by G ([8] Proposition 6.12), it follows that X + , Y + ∈ L 1 ∩ L 2 = L + 1 . Now let ξ = i Y i X dK, and note that Proposition 2.3 applied to both J 1 and J 2 implies that π 1/2 0,1 (ξ) = 0. Comparing against equation (3.2) we obtain (ω −1 I ∓ ω −1 J )(ξ) = 0. Therefore ι Y ι X dK = 0, for all X, Y ∈ T 1,0 I . Since K is real it follows that dK = 0. Proof of Theorem 1.1. Fix (J 1 , J 2 ) generalized Kähler and fix (J t 1 , J t 2 ) a one-parameter family as in the statement. First let us assume conditions (1), (2), and (3) hold for this family. Since dK t = 0 for all t, it follows from Corollary 2.4 that (J t 1 , J t 2 ) are integrable generalized complex structures. Furthermore, using that K t ∈ Λ 1,1 J t for all t, it follows from Lemma 2.2 and Proposition 3.3 that [J t 1 , J t 2 ] = 0 for all t. Since we have assumed the positivity of −J 1 J 2 ·, · in condition (3), it follows that (J t 1 , J t 2 ) is generalized Kähler for all t. Conversely, suppose (J t 1 , J t 2 ) defines a generalized Kähler structure for all t. Condition (3) then holds by definition, and condition (1) holds by Lemma 2.2 and Proposition 3.3. As the structures (J t 1 , J t 2 ) are assumed integrable for all times t, their Nijenhuis tensors vanish for all t, and thus it follows from Proposition 2.3 that 0 = π J i 0,1 dK π T π J i 1,0 ( x), π T π J i 1,0 ( y), · for all x, y ∈ T ⊕ T * , all t, and i = 1, 2. It then follows from Proposition 3.4 that dK = 0, as required.
With this characterization of canonical deformations in hand, we can now give the definition of generalized Kähler classes which emerges naturally from Theorem 1.1.
Definition 3.5. Let M be a smooth manifold. An exact canonical deformation is a one-parameter family of generalized Kähler structures (J t 1 , J t 2 ) such that, for all t, ∂ ∂t for some a t ∈ Λ 1 . Given J 1 , J 2 another generalized Kähler structure, we define an equivalence relation where if and only if there exists an exact canonical deformation (J t 1 , J t 2 ), t ∈ [0, 1], such that (J 0 1 , J 0 2 ) = (J 1 , J 2 ), (J 1 1 , J 1 2 ) = ( J 1 , J 2 ). Furthermore, the generalized Kähler class of (J 1 , J 2 ) is 3.5. Induced variations. In this section we derive the variation on the associated bihermitian data induced by a canonical deformation through an analysis of the Gualtieri map.
Proposition 3.6. Let (J t 1 , J t 2 ) be a canonical family, and let (g t , b t , I t , J t ) denote the corresponding 1-parameter family of bihermitian data. Theṅ Proof. We use the notation and computations of Proposition 3.3. In particular, we recall that .
On the other hand, writing expression 3.3 as J 1/2 = 1 2 e b Υ 1/2 e −b and differentiating, using the fact that e bėb =ė b =ė b e b , yields Then equating the appropriate expressions coming from above shows For the remaining data we differentiate the generalized metric where g t is the associated metric and b t = −g t A t . It follows that and in particular we will only be interested in the first row. Focusing on the top row, we furthermore compute where we have used that K ∈ Λ 1,1 J . Then differentiating the formulas I/J = −ω −1 I/J g giveṡ Example 3.7. Given (M 2n , g, J) a Kähler manifold, we interpret this as a generalized Kähler structure by setting I = J, and b = 0. Suppose (g t , b t , I t , J t ) is a canonical family with this initial condition. Initially we have σ = 0, thus σ t ≡ 0 from Corollary 1.3. It follows that [I, J] ≡ 0 for all times, so from the equations of Proposition 3.6 it follows that thatJ ≡ 0 for all times, and so J t ≡ J = 0. In turn it follows easily thatω I =ω J = K, in other words, the complex structures stay fixed and the Kähler forms change by K. By the √ −1∂∂-Lemma, an exact canonical deformation satisfies K = da = dJdu for some u ∈ C ∞ (M ). Using the construction above and the normalization that √ −1∂∂ = dId we verify item (1) of Proposition 1.2, noting that positivity of the Kähler forms (ω I ) t = ω I + tdJdu is equivalent to the positivity condition (3) in Theorem 1.1. Now suppose (g t , b t , I t , J t ) is a canonical family with this initial condition. Arguing as above, since σ = 0 for the given structure, it follows that [I, J] ≡ 0 for all times t, and henceJ = 0. Thus along the variation we preserve the splitting induced by Q, and we can decompose K = K ++ + K +− + K −+ + K −− . Tracing through the formulas in Proposition 3.6 yieldṡ A special case of this occurs when K = dJdu, yieldinġ where d = ∂ + + ∂ − + ∂ + + ∂ − is the fourfold splitting of d induced by Q. This construction verifies item (2) of Proposition 1.2, again noting that positivity of is equivalent to the positivity condition of item (3) in Theorem 1.1.
Example 3.9. Suppose (M 2n , g, b, I, J) is a generalized Kähler structure where the associated Poisson tensor σ is nondegenerate. In this setting the endomorphism [I, J] is invertible, and this in turn implies that I ± J are invertible. We define the 2-forms F ± = −2g(I ± J) −1 . Moreover let Ω = σ −1 . It turns out that the three symplectic forms F ± , Ω completely determine the generalized Kähler structure in this case. Direct computations show that the generalized complex structures can be expressed as Let K be an infinitessmal deformation of the generalized Kahler pair (J 1 , J 2 ). Then with respect to the data (F + , F − , Ω), direct computations show thaṫ F + =Ḟ − = K,Ω = 0.
As in the above examples we can choose K t = dJ t du t . We claim that for such a variation, To show this we compute, using that σ = Ω −1 and dΩ = 0, (L σdu J) Ω = L σdu (JΩ) − JL σdu Ω = dJdu = K.
Comparing against Proposition 3.6 we see thaṫ as required. It follows that J t = φ * t J 0 , where φ is the one-parameter family of Ω-Hamiltonian diffeomorphisms driven by u t . As discussed in [3], the positivity of − Im π Λ 1,1 I Ω J is equivalent to the positivity condition (3)

Generalized Kähler-Ricci flow as canonical deformation
In this section we establish Theorem 1.4, namely that solutions to the generalized Kähler-Ricci flow are canonical deformations, driven by the Bismut Ricci curvature. This generalizes and unifies various instances of this phenomenon which have previously been observed. In particular, it is well known that Kähler-Ricci flow moves within the Kähler cone against a fixed complex structure, and so is a canonical deformation as in Example 3.7. Next, comparing Example 3.8 against the curvature identities of ( [12]), we see that the generalized Kähler-Ricci flow in the case [I, J] = 0 is a canonical deformation driven by ρ I . Also, we can compare the discussion in Example 3.9 with ( [3]) to see the phenomenon holds in the nondegenerate case. The key point in establishing the general case is to show that the evolution of J is indeed determined by K = ρ I . This requires a delicate curvature identity we build up below.  Proof. We will use that for a Hermitian manifold (M 2n , g, J) one has g((L X J)Y, Z) = g((D X J)Y − D JY X + JD Y X, Z).