Blow-ups in generalized K\"ahler geometry

We continue the study of blow-ups in generalized complex geometry with the blow-up theory for generalized K\"ahler manifolds. The natural candidates for submanifolds to be blown-up are those which are generalized Poisson for one of the two generalized complex structures and can be blown up in a generalized complex manner. We show that the bi-Hermitian structure underlying the generalized K\"ahler pair lifts to a degenerate bi-Hermitian structure on this blow-up. Then, using a deformation procedure based on potentials in K\"ahler geometry, we identify two concrete situations in which one can deform the degenerate structure on the blow-up into a non-degenerate one. We end with an investigation of generalized K\"ahler Lie groups and give a concrete example on $(S^1)^n\times (S^3)^m$, for $n+m$ even.


Introduction
Generalized Kähler geometry was born in 1984 when Gates, Hull and Roĉek [10] discovered new supersymmetric sigma-models in physics, whose background geometry could be relaxed from Kähler to generalized Kähler. At that time it appeared in the guise of bi-Hermitian geometry; pairs of complex structures (I+, I−) compatible with a common metric g, satisfying an additional integrability equation. Bi-Hermitian geometry then gained interest from mathematicians and some theory was developed, notably in dimension four. For instance, a classification of four-manifolds carrying two compatible complex structures (I+, I−) with I+ = ±I− has been obtained in the cases where the metric is anti-self-dual (Pontecorvo [19]) and where the first Betti number is even (Apostolov, Gauduchon and Grantcharov [1]).
Around 2003 generalized Kähler geometry appeared in a different formulation out of the work of Gualtieri [14], in the context of generalized complex geometry. Generalized geometry is centered on the idea of replacing the tangent bundle of a manifold by the sum of its tangent and cotangent bundle. This creates enough room for merging both complex and symplectic 2 Generalized geometry 2

.1 Generalized complex geometry
Let M be a real 2n-dimensional manifold and H a closed real 3-form. In generalized geometry the tangent bundle is replaced by TM := T M ⊕ T * M . It is endowed with a natural pairing X + ξ, Y + η := 1 2 (ξ(Y ) + η(X)) and a bracket on its space of sections called the Courant bracket: It satisfies the Jacobi identity but is not skew-symmetric.
Definition 2.1. A generalized complex structure on (M, H) is a complex structure J on TM which is orthogonal with respect to , and whose (+i)-eigenbundle L ⊂ TM C is involutive 1 .
A Lagrangian, involutive subbundle L ⊂ TM C is also called a Dirac structure, and generalized complex structures correspond in a one-to-one fashion with Dirac structures L satisfying the additional non-degeneracy condition L ∩ L = 0. Dirac structures can conveniently be described in terms of differential forms. There is a natural action of the Clifford algebra of (TM, , ) on forms given by (X + ξ) · ρ = ι X ρ + ξ ∧ ρ, giving an identification between the space of differential forms and the space of spinors for Cl(TM, , ). A line subbundle K ⊂ Λ • T * M C gives rise to an isotropic subbundle L ⊂ TM C by taking the annihilator This yields to a one-to-one correspondence between Dirac structures L ⊂ TM C , and complex line bundles K ⊂ Λ • T * M C which satisfy the following two conditions. Firstly, K has to be generated by pure spinors, i.e. forms ρ which at each point x admit a decomposition ρx = e B+iω ∧ Ω (2.3) where B + iω is a 2-form and Ω is decomposable. This condition is equivalent to L being of maximal rank. Secondly, if ρ is a local section of K there should exist X + ξ ∈ Γ(TM C ) with This condition amounts to the involutivity of L. The condition L ∩L = 0 can then be expressed in spinor language using the Chevalley pairing: If ρ ∈ Γ(K) is non-vanishing then L ∩L = 0 ⇐⇒ (ρ,ρ) Ch := (ρ ∧ρ T )top = 0.
1 A subbundle of TM is called involutive if its space of sections is closed with respect to the Courant bracket.
Here the superscript T stands for transposition, acting on a degree l-form by (β1 . . . β l ) T = β l . . . β1, and the subscript top stands for the highest degree component. If ρ is given by (2.3) at a particular point x then this condition becomes ω n−k ∧ Ω ∧Ω = 0, (2.4) where k := deg(Ω). The line bundle K associated to a generalized complex structure is called the canonical line bundle, and the integer k appearing in (2.4) is called the type at x. Structures of type 0 and n are called symplectic, respectively complex 2 . Another description of the type is as follows. Every generalized complex structure naturally induces a Poisson structure given by the composition The conormal bundle to the leaves, i.e. the kernel of πJ , is given by the complex distribution In general νJ might be singular, for its complex dimension may jump in even steps from one point to the next. Intuitively, one can think of generalized complex structures as Poisson structures with transverse complex structures, and the type at a point x equals the number of complex directions, i.e.
We will often abbreviate (ϕ, 0) by ϕ and drop the prefix "generalized". An important role is played by B-field transformations, maps of the form 3 (Id, −B) =: e B * . They act on TM via Given u ∈ Γ(TM ) we denote by ad(u) : Γ(TM ) → Γ(TM ) the adjoint action with respect to the Courant bracket. This infinitesimal symmetry has a flow, i.e. a family of isomorphisms ψt : TM → TM with d/dt(ψt(v)) = − u, ψt(v) . If u = X + ξ and ϕt is the flow of X then where Bt := t 0 ϕ * r (dξ + ι X H)dr. A map Φ = (ϕ, B) gives rise to a correspondence: we say that X + ξ is Φ-related to Y + η, and write X If Φ is in addition invertible we call it an isomorphism.
Remark 2.5. It follows immediately from the definition that ϕ is a Poisson map, i.e. ϕ * πJ 1 = πJ 2 . This is quite restrictive, for example if the target is symplectic then ϕ has to be a submersion. In the complex category we recover the usual notion of holomorphic maps. In case ϕ is a diffeomorphism a more concrete description in terms of spinors can be given. If Ki is the canonical bundle for Ji, Φ being an isomorphism amounts to We can now state the analogue of the Newlander-Nirenberg and Darboux Theorems in generalized complex geometry. Theorem 2.6 ([2]). Let (M, H, J ) be a generalized complex manifold. If x ∈ M is a point where J has type k, then a neighborhood of x is isomorphic to a neighborhood of (0, 0) in where ωst is the standard symplectic form, σ is a holomorphic Poisson structure which vanishes at 0, and Jω st and J (i,σ) are defined in Example 2.2.
To blow up submanifolds we need an appropriate notion of generalized complex submanifold. The definition that we will use generalizes complex and symplectic submanifolds, as those are the submanifolds that are known to admit a blow up in their respective categories (there is for instance no blow-up available for (co-)isotropic submanifolds of a symplectic manifold). Let Φ = (ϕ, B) be a map and L2 a Dirac structure on (M2, H2). We define the backward image of L2 along Φ by This is a Dirac structure on (M1, H1), provided it is a smooth vector bundle. A sufficient condition for this is that ker(dϕ * ) ∩ ϕ * L is of constant rank. More information can be found in [4].
In complex or symplectic manifolds we recover the usual notion of complex respectively symplectic submanifolds. Also, a point is always a generalized complex submanifold. Note that if M is symplectic, i is generalized holomorphic if and only if Y is an open subset.

Generalized Kähler geometry
Generalized complex structures were introduced with the purpose of unifying complex and symplectic structures into one framework. On a Kähler manifold we have both a complex and a symplectic structure which are compatible with each other. Here is the generalized version. Since G 2 = 1, TM decomposes into its (±1)-eigenspaces V+ and V−, on which the pairing restricts to a positive, respectively negative definite form. So choosing a generalized metric amounts to choosing a reduction of structure groups for TM from O(2n, 2n) to O(2n)×O(2n). The restriction of J1 induces a complex structure on V±, leading to decompositions As J2 equals ±J1 on V±, we obtain Because V± does not intersect the isotropics T M and T * M , the projection V± → T M is an isomorphism and we can write V± as the graph of a map a± : T M → T * M . If we decompose a+ = g+b where g is symmetric and b is skew, then positivity of , |V + implies that g is positive definite, while orthogonality of V+ and V− implies a− = −g + b. Transporting the complex structure given by J1 on V± to T M we obtain two almost complex structures I± on M , both compatible with g. Such a tuple (g, b, I+, I−) will be called an almost 4 bi-Hermitian structure.
An explicit relation between (J1, J2) and (g, b, I±) is given by From this we see that It follows that πJ 1 + πJ 2 is invertible and, using a little bit of linear algebra, this implies that type(J1) + type(J2) ≤ n.
One can also relate the parity of the types of J1 and J2 to the orientations of I±. In general, on a 2n-dimensional manifold, type(J1) = n mod 2 if and only if I+ and I− induce the same orientation, while type(J2) = n mod 2 if and only if I+ and −I− induce the same orientations. Remark 2.12.
1 If (J1, J2) is generalized Kähler then so is (J2, J1), with the same generalized metric G. So when considering e.g. a generalized Poisson submanifold (c.f. Section 4.1) for one of the two structures, we may as well assume this to be J1.
2 We can always gauge away the two-form b by a transformation of the form (2.6), at the expense of modifying H by H − db. In what follows we will often implicitly assume this has been done, and we will refer to the tuple (g, I±, H) as the bi-Hermitian structure.
The difficult feature of bi-Hermitian geometry lies in the fact that I+ and I− do not commute in general. Therefore, standard techniques in complex geometry, such as decomposition of forms into types, become difficult as they can be performed only for one of the two complex structures at a time. This failure of commutativity suggests that important information about the generalized Kähler structure is contained in the tensor Since Q is skew-symmetric we can regard it as a bivector, and it was observed in [1] in the 4-dimensional case and in [15] in the general case, that Q is Poisson. In fact, it turns out to be the real part of two holomorphic Poisson structures σ± := Q − iI±Q. (2.14) One can prove this directly in local coordinates using the integrability conditions (see [15]), or in the following more abstract way (see [13]). If L1 is a Dirac structure on (M, H1) and L2 a Dirac structure on (M, H2), we can form their Baer-sum on (M, H1 + H2) via where i = (i, 0) : (M, H1 + H2) → (M, H1) × (M, H2) denotes the diagonal map, and the backward image is defined in (2.9). A sufficient condition for L1 L2 to be smooth is that L1 ∩ L2 ∩ T * M is of constant rank. Note that there is a natural map L1 ×T M L2 → L1 L2 given by (X +ξ, X +η) → X +ξ +η, which is an isomorphism if and only if L1 ∩L2 ∩T * M = 0. The latter condition also ensures that for spinors ρ1, ρ2 for L1, L2, the product ρ1 ∧ ρ2 does not vanish and forms a spinor for L1 L2.
Observe that T M , considered as Dirac structure on (M, 0), acts as a two-sided identity with respect to the Baer-sum operation, i.e. L T M = L = T M L. There is also an inverse for each Dirac structure L on (M, H), given by If L comes from a generalized complex structure, a quick computation shows that This is one way to see that πJ is integrable. In a similar spirit we have the following proposition which follows from the results in [13], and we give a proof for completeness.
Proof. We will only show that ; equality then follows from dimensional reasons and the case of σ− is similar. We have where T 1,0 + M denotes (+i)-eigenspace for I+. We can write X = X − g(X) + g(X) and since X +g(X) ∈ V 0,1 + = L1 ∩L2, we see that X ∈L T 1 L2. Next, let us denote by P± := 1 2 (1−iI±) the projections onto T 1,0 ± M . A quick calculation yields Using this we obtain, for ξ ∈ T * 1,0 The fact thatL T 1 L2 is smooth can also be seen directly fromL where Ω is a suitably scaled (n, 0)-form for I+. Similarly, We conclude this section with a bit of linear algebra that will be needed later. . On this subspace I+ and I− commute with each other and so they admit a simultaneous eigenspace decomposition, all of whose eigenvalues are ±i. The result follows.

A flow of bi-Hermitian structures
In the process of blowing up generalized Kähler manifolds one encounters metrics which degenerate along the exceptional divisor. To deal with this we introduce a deformation procedure to flow a degenerate structure into a non-degenerate one. The idea behind this flow already appears in [16] where it was used to describe new examples of generalized Kähler manifolds, and it was subsequently used in [8] for the blow-up procedure. The following definition is intended to capture the situation encountered in the blow-up.  3. The terminology originates from the situation where α = −df for a densely defined function f , which is usually referred to as the potential. Although this is the situation in which we are interested, we state the results in this section for general closed 1-forms. Let α be a potential for I+. Denote by ϕt the flow of Xα and define closed 2-forms We will have to be careful with G − t and F − t , since d c − α is not assumed to be smooth. The aim of this section is to prove Theorem 3.4. Let (g, I+, I−, H) be a degenerate bi-Hermitian structure with compact degeneracy submanifold E, and let α be a potential for I+ such that d c + α has compact support and Then the tuple (gt, I+,t, I−,t, Ht), where forms a bi-Hermitian structure for sufficiently small t > 0.
Proof. It is clear that gt is symmetric, I±,t are integrable and that Ht is closed for all t. By construction gt is compatible with I−,t = I−. Let us show that gt is a metric for sufficiently small t > 0. Choose a relatively compact open neighborhood V of E in M with supp(d c + α) ⊂ V and pick a δ1 > 0 and a relatively compact open set W with ϕt(V ) ⊂ W for all t ≤ δ1. By construction F + t = 0 on M \W for t ≤ δ1 and so gt = g is non-degenerate there. Writing gt = g + ht, then as t goes to 0, ht/t converges toḣ0 which by assumption is positive on T M ⊥ . For small > 0, g + ḣ 0 is positive on T M |E and therefore also on T M | U for U ⊂ W a small enough neighborhood of E. Hence the same is true for g + ht/t, and therefore also for tg/ + ht, provided t > 0 is close to zero. If in addition t ≤ then tg/ + ht ≤ g + ht = gt since g ≥ 0. In conclusion, there exist a neighborhood U of E in W and a δ2 > 0 such that gt is positive on U for all 0 < t ≤ δ2. Since W \U is compact and g0 = g is non-degenerate on M \E there is a δ3 > 0 such that gt is positive on W \U for 0 ≤ t ≤ δ3. Consequently, gt is a metric for 0 < t ≤ min(δ1, δ2, δ3). As already observed above, gt is compatible with I−,t for all t. Moreover, since any closed 2-form F on a complex manifold satisfies Hence all that is left to verify is that I+,t is also compatible with gt and that d c +,t ω+,t = Ht. In contrast with the I−,t case this is not immediately obvious, the reason being that the flow seems to treat I+ and I− on an unequal footing. However, we will now show that if we pull back (3.1) by ϕt, we obtain a similar flow but with the roles of I+ and I− interchanged. We begin by giving an alternative formula for I+,t.
Proof. Consider the generalized complex structure integrable with respect to the zero 3-form (c.f. Example 2.2). As α is closed, the generalized vector field J+α = Xα − I * + α is a symmetry, which means that J+ is preserved by its flow 5 = ϕt * J+ and in particular ϕt * I+ = I+ − QF + t . For I− we have to be careful since we do not know whether F − t is smooth. We can apply the above argument on the open dense set where α is smooth, if we keep the time parameter small. In particular, we conclude that LX α I− = Qd c − α holds on the dense set where α is defined. Since Xα is smooth, we learn that Qd c − α, and therefore also QG − t and QF − t , are smooth. The statement for I− then follows because it holds at t = 0, and because both sides have equal time derivatives.
In the lemma below we denote by ∇ the Levi-Cevita connection and by ∇ ± the connections defined in Proposition 2.11, which are defined on the open set M \E where g is a metric.
Proof. We will verify these expressions on the intersection of M \E with the open dense set where α is defined. Since the left-hand sides of both equations are smooth, this shows that the right-hand sides have smooth extensions over all of M . As both sides of (3.2) are symmetric, it suffices to evaluate them on a pair (Y, Y ). Using 5 Although J + α is only densely defined, its associated adjoint action on Γ(TM ) depends only on Xα and d c + α and is therefore defined everywhere. In particular, the flow is also defined everywhere.

proving (3.3).
From the proof of Lemma 3.5 we learned that Qd c − α is smooth, and therefore also d c − αQ by taking adjoints. Equation (3.2) gives us in addition smoothness of (d c − α) 1,1 I + , and if we apply d to Equation (3.3) we see (after some rearranging) that d(I * is also smooth. Combining this with Lemma 3.5, we conclude that both I * which is what we need to make sense of the following lemma.
Proof. Both equations hold at t = 0, so it suffices to show that both sides have the same . Using Lemma 3.5 and Equation (3.2) we obtain This equals the time derivative of the right-hand side, thereby proving (3.6). For (3.7), using 6 Here I * ± denotes the action on forms which on a form of degree (p, q) acts by i(p − q). 7 Here we use the notation (+ ↔ −) to denote the same term that precedes it but with ± symbols interchanged.
We can rewrite (3.7) as and since the first term is closed, we obtain which equals the time derivative of the right-hand side of (3.7), thereby proving it.
From Lemma 3.7, together with the arguments applied before to I−,t, it follows that ϕ * t I+,t = I+ is compatible with ϕ * t gt, and that Pushing everything forward again by ϕt we obtain the desired compatibility of I+,t with gt and Ht, finishing the proof of Theorem 3.4.
Remark 3.8. We stated the theorem for potentials for I+ but of course a similar result is true for potentials for I−. In that case we need I * + (−d c − α) 1,1 I + to be positive on T M ⊥ . Remark 3.9 ([16]). Theorem 3.4 is stated for metrics which are almost everywhere nondegenerate. It can however, also be applied to the following situation where g is identically zero. Suppose (M, I) is a compact complex manifold and σ a holomorphic Poisson structure with real part Q. Setting g = 0, I± = I and H = 0, this is a degenerate bi-Hermitian structure in the sense of Definition 3.1, except for the fact that the degeneracy set E = M is no longer nowhere dense. Still, Lemma 3.5 only used that Q is the real part of holomorphic Poisson structures σ± while Lemma 3.6 is trivially true in this case (both sides of both equations are zero). Hence Theorem 3.4 still applies in this case, and all we need is a potential α such that −d c α is positive on M . To that end, suppose that D ⊂ M is a divisor which is Poisson for σ and which in addition is positive, i.e. the line bundle OX (D) is positive. Let s ∈ Γ(OX (D)) be a holomorphic section which vanishes to first order along D and choose a Hermitian metric h on OX (D) such that iR h is positive, where R h is the curvature of the unitary connection induced by h. Then α := −d log |s| is a potential with the desired properties, for −d c α = iR h is smooth and positive, while Q(α) is smooth because D is Poisson (c.f. the proof of Theorem 4.5(i)). The deformation procedure then gives us a bi-Hermitian structure where I+ and I− are no longer equal. In [16] this was applied to find examples of generalized Kähler structures on Del Pezzo surfaces, which are complex surfaces whose anticanonical bundle is positive.

Blowing up submanifolds 4.1 Blow-ups in generalized complex geometry
We briefly summarize the results on blowing up submanifolds in generalized complex geometry as treated in [3]. The most general notion of blowing up a submanifold Y ⊂ M involves the notion of holomorphic ideal; an ideal IY ⊂ C ∞ (M ) of complex valued smooth functions with Y as its zero set, and which is locally around a point in Y generated by functions z 1 , . . . , z l that form a submersion to C l . The blow-up π :M → M of Y in M is then defined by the following universal property: for any map f : X → M for which f * IY is a divisor 8 there is a unique factorization of f through π. It is shown in [3] that for every holomorphic ideal there exists a (canonical) blow-up.
There are two kinds of generalized complex submanifolds 9 that admit a blow-up. Firstly, we have the generalized Poisson submanifolds, by definition these are the submanifolds Y ⊂ M with J N * Y = N * Y , and they behave in a complex manner in normal directions. It turns out that these submanifolds inherit a canonical holomorphic ideal from the generalized complex structure, and so there is a canonical (differentiable) blow-up. This is however not automatically generalized complex. To phrase the precise condition, observe that a generalized Poisson submanifold is in particular Poisson for πJ , and so N * Y inherits a fiberwise Lie algebra structure. Concretely, if α, β ∈ N * y Y and X ∈ TyM , we have where X is an arbitrary local extension of X. This Lie bracket turns out to be complex linear, where the complex structure on N * Y is given by the restriction of J . We call N * Y degenerate if the bundle map vanishes. Here both the exterior and symmetric algebra are taken over C, which is well-defined since N * Y is complex. Note that degeneracy is really a property of Lie algebras, and N * Y being degenerate means that all its fibers are. Another description of degeneracy for a Lie algebra is that the bracket of any two elements lies in the plane spanned by them. Then, the differentiable blow-up of Y in M admits a generalized complex structure for which the blow-down map is holomorphic if and only if N * Y is degenerate.
The second class of submanifolds is formed by the generalized Poisson transversals, defined by the condition J N * Y ∩ (N * Y ) ⊥ = 0. They behave symplectically in normal directions and admit a global neighborhood theorem. If in addition Y is compact, using the neighborhood theorem one can construct a (non-canonical) generalized complex blow-up. To put this blow-up in the context of holomorphic ideals, observe that a submanifold Y equipped with a complex structure on its normal bundle inherits many holomorphic ideals by taking the canonical ideal of fiberwise linear functions on the manifold N Y , and transporting it into M using a tubular neighborhood. Then the blow-up of a generalized Poisson transversal can be regarded as the blow-up with respect to such an ideal. Since it is non-canonical, the ideal description is not really useful in this context. On a generalized Kähler manifold the two types of submanifolds are related as follows. Proof. By the generalized Kähler condition we have In particular, if Y is a generalized Poisson submanifold for J1 such that N * Y is degenerate, then Y can be blown up for both J1 and J2, albeit in different manners. It sounds reasonable to expect that there will be a generalized Kähler blow-up for Y in M . Although a complete answer is still lacking, in the next sections we will give some extra sufficient conditions that will guarantee the existence of a generalized Kähler blow-up. The strategy will be the following: Given a Y ⊂ (M, J1, J2) as above, we blow it up for J1 and then show that the bi-Hermitian structure lifts to a degenerate bi-Hermitian structure on the blow-up. Then, under additional hypotheses we apply the flow procedure of the previous section which makes the structure non-degenerate, and the result will be the desired blow-up.

Lifting the bi-Hermitian structure
Let Y ⊂ (M, J1, J2) be a generalized Poisson transversal for J1. Associated to (J1, J2) there is the pair of holomorphic Poisson structures (I±, σ±) defined in (2.14), whose real parts equal Q. We want to show that (I±, σ±) all lift to the generalized complex blow-up of Y with respect to J1. To that end we will first prove that σ± lifts to the blow-up of Y with respect to I±, and then show that the three different blow-ups coincide.
From ( for α, β ∈ N * y Y , whereα,β are smooth local extensions of α and β. Abstractly, if (g, [, ]) is a Lie algebra and A : g → g a linear map such that [u, v]A := [Au, v] is again a Lie bracket 10 , then degeneracy of [, ] implies that for [, ]A as well. To prove this, we need to show that [x, y]A ∈ C · x + C · y for all x, y ∈ g. It suffices to verify this for x, y that are linearly independent. Since [, ] is degenerate, there are λ, µ ∈ C with [x, y]A = [Ax, y] = λAx + µy. Since [, ]A is skew, we have [Ax, y] = [Ax, x + y], and so there are λ , µ ∈ C with [x, y]A = λ Ax + µ (x + y). Comparing both equations, we see that either Ax ∈ C · x + C · y and hence also [Ax, y] ∈ C · x + C · y, or [Ax, y] = λAx. Running the same argument with x and y interchanged we see that either [Ax, y] ∈ C · x + C · y, or [Ax, y] = µAy for some µ ∈ C. If [Ax, y] is nonzero, Ax is proportional to Ay and so [Ax, y] = 0 by skew symmetry of [, ]A, a contradiction. Hence, [x, y]A ∈ C · x + C · y ∀x, y ∈ g, and so [, ]A is degenerate.
By a result of Polishchuk [18] it follows that Y can be blown up for both (I±, σ±). Let us denote byM the blow-up with respect to J1 and byM± the blow-up with respect to I±. Proof. As discussed in Section 4.1 and more precisely explained in [3], the blow-upM is constructed from a holomorphic ideal IY,J 1 that Y inherits from J1, while the blow-upsM± use the natural holomorphic ideals IY,I ± that Y inherits from being a complex submanifold of (M, I±). It thus suffices to show that these three ideals coincide, which turns out to be true up to a conjugation, i.e. IY,J 1 = IY,I ± . This is not a problem, for the blow-up of a conjugate ideal is given by the same manifold but with conjugate divisor. Pick a local chart (R 2n−2k , ωst) × (C k , σ) for J1 as provided by Theorem 2.6, in which Y necessarily looks like W ×Z where W ⊂ R 2n−2k is open and Z ⊂ C k is holomorphic Poisson. If (x, z) are coordinates on this chart in which Y = {z 1 , . . . , z l = 0}, it is shown in [3] that the ideal z 1 , . . . , z l is independent of the choice of local chart, so they glue together to form (by definition) the ideal IY,J 1 . Let us verify that IY,J 1 = IY,I + , the case of I− being similar. Pick a holomorphic chart for I+ with coordinates u i so that Y is given by {u 1 , . . . , u l = 0} and so IY,I + = u 1 , . . . , u l . As is proven for instance in [17], we can verify IY,J 1 ⊂ IY,I + merely by looking at Taylor series, i.e. we need to show that For this we use (2.17), which now explicitly becomes where e f is some rescaling. One can prove (4.3) by induction on m and by applying appropriate Lie derivatives to (4.4). For precise details on this part of the argument we refer to the proof of Proposition 2.6 in [3], which is almost identical. Hence IY,J 1 ⊂ IY,I + , and since both are holomorphic ideals for Y , we obtain IY,J 1 = IY,I + , which is what we needed to show.

Flowing towards a non-degenerate structure
Let Y ⊂ (M, J1, J2) be a generalized Poisson transversal for J1 with degenerate normal bundle, and denote by π :M → M the corresponding blow-up. By Lemma 4.2 we know that the complex structures I± lift toM , and together with π * g, π * H and the lift of Q they form a degenerate bi-Hermitian structure onM . The metric degenerates along the exceptional divisor E = π −1 (Y ), and TM ⊥ equals the vertical tangent bundle of the fibration π : E ∼ = P(N Y ) → Y . In order to apply the deformation procedure from Section 3 we need a suitable potential 11 f . This will be based on the following idea (see also [8]). Consider M andM as complex manifolds with respect to either I+ or I−, and E as a divisor oñ M with associated holomorphic line bundle OM (−E). Recall that a Hermitian metric on a holomorphic line bundle induces a unitary connection, whose curvature R h is of type (1, 1).
If U is any neighborhood of E inM , there exists a metric h on OM (−E) such that iR h is supported in U and restricts to a positive (1, 1)-form on TM ⊥ .
Proof. On E we have the tautological line bundle OE(−1) ⊂ π * N Y whose fiber over a point l ∈ E = P(N Y ) is the corresponding line in N Y . If we equip N Y with a Hermitian metric then this induces one on OE(−1) and therefore also on OE(1) := OE(−1) * . Denote the latter by h and its curvature by R h . If we set Ey := π −1 (y) = P(NyY ) for y ∈ Y , then iR h |E y equals (a multiple of) the Fubini-Study form 12 on P(NyY ). Now OM (−E)|E ∼ = N * E, the conormal bundle of E inM , which in turn equals OE(1). We can extend the metric h on OE(1) to a metric on OM (−E) as follows. Forgetting about the holomorphic structure for a moment, pick a tubular neighborhood p : V → E with V ⊂ U , and use it to identify OM (−E)|V ∼ = p * OE(1). Equip OM (−E)|V with the metric p * h with curvature p * R h , which has the same restriction to all the Ey's as R h does. On the complement of E the bundle OM (−E) is trivial so can be given a flat metric h . We let h be equal to h onM \U , p * h on a neighborhood of E in V and a suitable interpolation in between. Clearly R h is compactly supported in U and iR h |E y is positive.
It is a well-known fact that if s is any meromorphic section of OM (−E) which is not identically zero, then iR h = −dd c log |s|. So f := − log |s| is a good candidate for a potential. Unfortunately, we can not always guarantee smoothness of the Hamiltonian vector field Q(df ). The theorem below gives two situations where Q(df ) can be controlled. Moreover, in situation i) the generalized Kähler structure on the blow-up agrees with the original structure on the complement of a neighborhood of the exceptional divisor. In situation ii) the same is true, if we make the additional assumption that OM (D)|Y is trivial. In that case it is also not necessary to assume that D is compact.
Proof. i): Consider M as a complex manifold with respect to, say, I+. From (4.2) we see that [, ]Q is Abelian on N * Y and therefore E ⊂M is a Poisson submanifold for the lift of Q. Consider the potential f = − log |s|, where s is a meromorphic section of OM (−E) with a simple pole along E, and the norm is taken with respect to a metric as in Lemma 4.4. We claim that Q(df ) extends smoothly to the whole ofM . To see this, let x ∈ E and let e be a local holomorphic section of OM (−E) with e(x) = 0. Then s = 1 z e, where z is a local equation for E, hence Since σ+(dz) vanishes on E, it is divisible by z and we see that Q(df ) is indeed smooth. We already know that dd c + f is smooth and that dd c + f | TM ⊥ is positive, but in order to apply Theorem 3.4 we need that (I * − (dd c + f ) 1,1 I − )| TM ⊥ is positive. However, the complex structure on Ey is induced from NyY under the isomorphism Ey = P(NyY ). Since Y is generalized Poisson, both I+ and I− coincide on N Y and preserve it. So Ey is a complex submanifold of M with respect to both I+ and I−, with the same induced complex structure. In particular (I * − (dd c + f ) 1,1 I − )|E y = dd c + f |E y is positive. So Theorem 3.4 applies and we obtain a generalized Kähler structure by perturbing the structure in a neighborhood of E, whose size is controlled by the choice of metric in Lemma 4.4 (so in particular can be arbitrarily small). ii): LetD denote the proper transform 13 of D on the blow-upM . In terms of divisors, D = π * D−kE for some k ∈ Z>0 and so OM (D) = OM (−kE)⊗π * OM (D). Equip OM (−kE) = OM (−E) ⊗k with the metric h ⊗k , where h is a metric on OM (−E) as in Lemma 4.4. If h is any metric on OM (D), the metric h ⊗k ⊗ π * h on OM (D) satisfies iR h ⊗k ⊗π * h = ikR h + iπ * R h , which is positive on TM ⊥ since π * R h vanishes there. Let s be a holomorphic section of OM (D) with a simple zero alongD, and define f := − log |s|. Then dd c f = ikR h + iπ * R h is smooth onM and positive on TM ⊥ , while the same argument as in i) shows that Q(df ) is smooth, using the fact thatD is Poisson. So again Theorem 3.4 applies, but this time the structure is perturbed alongD as well so we can not contain the deformation to a neighborhood of E. If however we know that OM (D)|Y is trivial, then we can choose h above to be flat around Y and so dd c + f = ikR h around E. If s is a section of OM (kE) with a zero of order k along E and OM (kE) is equipped with the metric dual to h ⊗k , we can define f := −ρ·log |s |, where ρ is a function which is 0 near E and 1 outside of a neighborhood of E. Then f + f still has the property that Q(df ) is smooth but in addition satisfies dd c (f + f ) = 0 on an annulus around E. We then apply the deformation procedure only on a neighborhood of E, keeping it fixed on an annulus around it, and then glue the result back to the original structure.
Remark 4.6. i): Suppose that M is a Kähler manifold, seen as a generalized Kähler manifold as in Example 2.10, and Y is a complex submanifold regarded as a generalized Poisson submanifold for J1. Then, since πJ 1 = 0, N * Y is Abelian and we are in situation i) of the Theorem. Equation (3.1) that defines the flow reduces in this equation to simply adding dd c f to the symplectic form, and this is how one usually produces a Kähler metric on the blow-up. ii): Let us clarify why we need OM (D)|Y to be trivial if we want to contain the deformation to a neighborhood of E in situation ii). In the first part of the proof we are flowing the structure by the 2-form ikR h + iπ * R h , and in the second part we want to cancel this on an annulus around E by dd c f , where f is a smooth function. In particular we need ikR h + iπ * R h to be exact on the annulus, which is automatic for ikR h since OM (kE)|M \E is trivial. For π * R h to be exact, we need R h to be exact 14 around Y , which amounts to OM (D)|Y being trivial around Y . Although condition ii) of the theorem is clear as it is stated, it is unclear whether it has any applications. For that reason we state the following Proof. Let X1 be the type change locus for J1. In a local chart of the form (2.8), X1 is given by the vanishing of the holomorphic function σ k/2 and as such is either empty or a codimension 1 analytic subset of C n . We assume X1 = ∅, otherwise the statement is vacuous. Let D ⊂ M denote the Poisson subvariety of points where Q does not assume its maximal rank on M . By Lemma 2.14 and Equation (2.12) we have ker(Q) = ker(I * + − I * − ) ⊕ ker(I * + + I * − ) = ker(πJ 1 ) ⊕ ker(πJ 2 ).
Consequently, D = X1 ∪ X2, where Xi is the set of points where πJ i is not of maximal rank (or equivalently, where Ji is not of minimal type). Let D be the union of the codimension 1 components of D . Then D is also a Poisson subvariety which a priori could be empty, but we claim that X1 ⊂ D. Indeed, if x ∈ X1\D, then a neighborhood U of x in X1 is disjoint from D. However, since U is given by the vanishing of a holomorphic function (for a complex structure which need not coincide with either I±), an open dense set in U is a smooth submanifold of M of real dimension 2n − 2. But U ⊂ D , and a real 2n − 2 dimensional submanifold of a complex manifold can not be contained in a finite union of analytic subsets of complex codimension bigger than 1. So indeed X1 ⊂ D and Theorem 4.5 ii) applies. 13 By definition this is the closure of π −1 (D)\E inM . 14 As Y has complex codimension 2 or bigger (otherwise the blow-up is trivial), the Gysin sequence shows that the second degree cohomology of an annulus around Y agrees with that of Y itself. 15 Since J 1 is generically symplectic, for Y to be generalized Poisson it has to be either an open set in the symplectic locus or fully contained in the type change locus. In the former case there is nothing to blow up.
A special case of this corollary is when Y is a point and M is 4-dimensional. This situation was considered in [8], where it was assumed that the type change locus was smooth at the point in question. Note that if the type change locus is not smooth at this point, we are in situation i) of Theorem 4.5 and we can still blow it up. Remark 4.8. In [11] Goto proved that every compact Kähler manifold with a holomorphic Poisson structure σ has a generalized Kähler structure, where J1 is given by the Poisson deformation of the complex structure and J2 is a suitable deformation of the symplectic structure. If Y is a complex Poisson submanifold for σ whose conormal bundle is degenerate, then σ lifts to the complex blow-up of Y , which has a Kähler metric of its own(cf. Remark 4.6 i)). Applying Goto's theorem again, we see that the blow-up is again generalized Kähler and the blow-down map is generalized holomorphic with respect to J1. This example shows that Theorem 4.5 is not optimal in its assumptions. However, the proof of Goto's theorem relies on the use of Green's operators for finding the right deformation of J2, and as such is non-local. In fact, since the Kähler metric on the blow-up differs from the original metric on a neighborhood of the exceptional divisor, there is a-priori nothing we can say about the relation between J2 on the blow-up and on the original manifold, even far away from the exceptional divisor. As such, it is not clear how to connect this example with the techniques used to prove Theorem 4.5.

An example: compact Lie groups
One source of examples of generalized Kähler manifolds is provided by Lie theory.
Proposition 5.1 ([13]). Let G be an even dimensional compact Lie group. Then G has a generalized Kähler structure.
In order to find suitable submanifolds to blow up, we need to understand these generalized Kähler structures in some detail. Let G be any Lie group. An element ξ ∈ g defines left and right invariant vector fields ξ L g := deLg(ξ) and ξ R g := deRg(ξ), and we have [ξ L , η L ] = [ξ, η] L , [ξ R , η R ] = −[ξ, η] R . These two trivializations of T G define connections ∇ ± , characterized by ∇ + ξ L = 0 = ∇ − ξ R for ξ ∈ g. Their torsion is given by Suppose now that G is compact and let , be a metric on g which is invariant under the adjoint action. In particular, its left and right invariant extensions over G coincide and we denote this common extension by the same symbol , . The 3-form on g defined by H(ξ, η, ζ) := [ξ, η], ζ is also invariant under the adjoint action and so extends to a bi-invariant 3-form on G. From the Jacobi identity it follows that H is closed 16 and we have T ± (X, Y ), Z = ∓H(X, Y, Z), so ∇ ± coincide with the connections defined in Proposition 2.11. Since G is compact it is automatically reductive, i.e. its Lie algebra splits as g = a ⊕ g , with a Abelian and g semi-simple. Let t be a maximal Cartan subalgebra of g and g C = t C ⊕ α∈R g α the associated root space decomposition. Since G is compact the roots are contained in it ⊂ t * C , hence they come in pairs ±α and we have g α = g −α . Consequently, dim(g ) and dim(t) have the same parity and since g is even dimensional it follows that a ⊕ t is even dimensional. Now choose a decomposition R = R − ∪R + into positive and negative roots, so that in particular −R+ = R−. We define a complex structure I on g by the decomposition g C = g 1,0 ⊕ g 0,1 , where and (a ⊕ t) 1,0 is defined with respect to an arbitrary but fixed complex structure on a ⊕ t, compatible with , . By invariance of the metric, gα is orthogonal to g β unless α = −β, hence I is compatible with , . Since the sum of two positive roots is again positive it follows that [g 1,0 , g 1,0 ] ⊂ g 1,0 , so the complex structures I+ and I−, which are defined by the left respectively right invariant extensions of I over G, are integrable. Since they are constant in the two respective trivializations, we have ∇ ± I± = 0, so (G, , , I±, H) is generalized Kähler by Proposition 2.11.
One readily verifies that this algebra coincides with a ⊕ t, hence the connected component of the complex locus that contains the identity equals the connected subgroup T whose Lie algebra is a ⊕ t. Thus T , or any complex submanifold Y ⊂ T for that matter, is a generalized Poisson submanifold of (G, J1). To blow up Y in G with respect to J1, we need to understand the induced Lie algebra structure on N * Y . Since Y ⊂ T , we have an inclusion of Lie algebras N * T |Y ⊂ N * Y ⊂ T * G|Y . (5.1) The action of T on G, either from the left or the right, is a symmetry of the whole generalized Kähler structure that preserves T . In particular, the Lie brackets on T * y G and N * y T are independent of y ∈ Y and we can compute them at e ∈ G. From (2.12) we see that where ω(ξ, η) = Iξ, η is the associated Hermitian two-form on g. Consequently, for ζ ∈ g. Let ξ, η ∈ g and denote byξ,η ∈ g * their images under the metric. We have [ξ,η]π J 1 (ζ) = (L ζ L πJ 1 )e(ξ,η) = 1 2 [Iξ, η] + [ξ, Iη], ζ .
for all ζ ∈ g. Hence, using the metric, the bracket [, ]π J 1 induces the following bracket on g: Write ξ = ξ + α (ξα + ξα), where ξ ∈ a ⊕ t and α (ξα + ξα) ∈ α (gα ⊕ gα) R , and similarly for η. Here and below, the summation on α is over all positive roots. Then Here we are regarding the roots α, β ∈ (a C ⊕ t C ) * by extending them trivially over a C , and define their (1, 0) and (0, 1) components with respect to I|a⊕t. From (5.1) we see that if N * y Y is degenerate then so is N * y T , and therefore the whole of N * T by T -equivariance. So a necessary condition to blow up anything in T is that T itself can be blown up. This can be seen by restricting (5.2) to (TeT ) ⊥ = α (gα ⊕ gα) R ⊂ g. There (5.2) reduces to Recall that a Lie algebra is degenerate if and only if the bracket of any two elements lies in the plane spanned by them. In particular, as [gα, g β ] = g α+β for root decompositions, N * e T ∼ = (TeT ) ⊥ is degenerate if and only if the sum of any two positive roots is not a root itself. The only root systems satisfying this are products of A1, corresponding to the Lie group SU (2). In conclusion, in order to blow up T in G with respect to J1, we need G to be of the form G = (S 1 ) n × (S 3 ) m for n + m even, with T = (S 1 ) n × (S 1 ) m . We then still have some residual freedom in choosing the complex submanifold Y ⊂ T . Instead of determining precisely all Y for which (5.2) becomes degenerate let us give an easy example. If m is even we can take the complex structure I on g to be a product complex structure, i.e. we can assume that I preserves the decomposition a ⊕ t. Then if Y is of the form Y × (S 1 ) m with Y ⊂ (S 1 ) n a complex submanifold, (5.2) vanishes on N * Y , because all roots vanish on a C . If m is odd, we can choose the complex structure to be a product on (S 1 ) n−1 × (S 1 × (S 3 ) m ). Then if Y = Y × (S 1 ) 1+m with Y ⊂ (S 1 ) n−1 complex, (5.2) again vanishes on N * Y . Note that since in all these cases N * Y is Abelian, Theorem 4.5 i) applies and we obtain Theorem 5.2. Let G = (S 1 ) n × (S 3 ) m , where n + m is even, and let T = (S 1 ) n × (S 1 ) m ⊂ G be a maximal torus. Equip G with a generalized Kähler structure as above for which T is generalized Poisson. If either i) m is even and Y = Y × (S 1 ) m with Y ⊂ (S 1 ) n complex, or ii) m is odd and Y = Y × (S 1 ) m+1 with Y ⊂ (S 1 ) n−1 complex, then Y can be blown up to a generalized Kähler manifold.