C^{\infty}-logarithmic transformations and generalized complex structures

# C∞-logarithmic transformations and generalized complex structures

Ryushi Goto Department of Mathematics, Graduate School of Science, Osaka University, Toyonaka, Osaka 560-0043, Japan  and  Kenta Hayano Department of Mathematics, Graduate School of Science, Osaka University, Toyonaka, Osaka 560-0043, Japan
###### Abstract.

We provide a new construction of generalized complex manifolds by every logarithmic transformations. Applying a technique of broken Lefschetz fibrations, we obtain twisted generalized complex structures with arbitrary large numbers of connected components of type changing loci on the manifold which is obtained from a symplectic manifold by logarithmic transformations of multiplicity on a symplectic -torus with trivial normal bundle. Elliptic surfaces with non-zero euler characteristic and the connected sums , and admit twisted generalized complex structures with type changing loci for arbitrary large .

###### Key words and phrases:
generalized complex structure; logarithmic transformation; symplectic structure; -manifolds
###### 2010 Mathematics Subject Classification:
Primary 53D18, 53C15; Secondary 53D05, 57D50

## 1. Introduction

Generalized complex structures are geometric structures on a manifold introduced by Hitchin which are hybrids of ordinary complex structures and symplectic structures [Hitchin_2003]. On a -manifold a generalized complex structure of type number is induced from a complex structure and the one of type number is obtained from a symplectic structure modulo the action of -fields. Type changing loci of a generalized complex -manifold is the set of points on which the type number of changes from to . We denote by the number of connected components of type changing loci of

Cavalcanti and Gualtieri [Cavalcanti_Gualtieri_2006, Cabalcanti_Gualtieri_2009] constructed intriguing generalized complex structures with on the connected sums for , which do not admit any complex and symplectic structures if . Torres [Torres_2012] constructed many examples of generalized complex structures with multiple type changing loci. They used logarithmic transformations of multiplicity to obtain generalized complex -manifolds.

In this note, we generalize their construction to arbitrary logarithmic transformations at first. A -torus on a generalized complex -manifold is called symplectic if is induced from a symplectic structure 111 A symplectic structure induces the generalized complex structure . on a neighborhood of on which is symplectic with respect to . We start with a generalized complex manifold with which has symplectic -tori of self-intersection number zero. Then it is shown that a nontrivial logarithmic transformation on the symplectic -tori of yields a new generalized complex structure with . We apply suitable logarithmic transformations of multiplicity successively which do not change the diffeomorphism type of a manifold. Thus we obtain a family of generalized complex structures with for every on a manifold.

If a generalized complex structure gives a generalized Kähler -manifold, it turns out that the number of connected components of type changing loci are less than or equal to two. In fact, generalized Kähler structures are equivalent to bihermitian structures and the type changing loci are given by the set which coincides with zeros of a holomorphic Poisson structures [A.G.G_2006, Hitchin_2007, Goto_2010]. Thus our result is a sharp contrast to the one of generalized complex structures in generalized Kähler manifolds.

In Section , we give a brief explanation of generalized complex structures. In Section , we give our construction of generalized complex manifolds (Theorem 3.1). Moishezon proves that logarithmic transformations of multiplicity one do not change the diffeomorphism type of a genus -Lefschetz fibration with a cusp neighborhood. Then Theorem 3.1 yields generalized complex structures with for every on a genus -Lefschetz fibration with a cusp neighborhood. In Section 4.1, we also show that the connected sum for admits generalized complex structures with for every . In Section 4.2, we develop a technique of broken Lefschetz fibrations222 Broken Lefschetz fibrations are introduced by Auroux, Donaldson and Katzarkov from the context of degenerate symplectic structures which are studied from view points of topology of -manifolds [A.D.K_2005] and show that a certain logarithmic transformation of multiplicity one on a -torus also preserves the diffeomorphism type of a manifold. This result has an independent importance from view points of topology of -manifolds (see Lemma 4.4). We apply Lemma 4.4 to a generalized complex -manifold with type changing loci and a symplectic -torus of self-intersection number zero. Then it turns out that the manifold which is given by logarithmic transformations of multiplicity on along admits a (twisted) generalized complex structure with type changing loci for every (Theorem LABEL:thm_GCS_multiplicity0). We apply Theorem LABEL:thm_GCS_multiplicity0 to the connected sum and (Corollary LABEL:cor_GCS_product_sphere and Corollary LABEL:cor_GCS_S1S3).

After the authors submitted their paper to the Arxiv, Rafael Torres kindly sent us a message that he and Yazinski constructed several examples of generalized complex structures with an arbitrary large type changing loci by a different method applying only logarithmic transformations of multiplicity [Torres_Yazinski].

## 2. Generalized complex structures

Let be a manifold of dimension with a closed -form . The -twisted Courant bracket of is given by

 [V+ξ,W+η]H=[V,W]+LVη−LWξ−12d(η(V)−ξ(W))+iViWH,

where denotes the ordinary bracket of vector fields and and is the Lie derivative of a -form by and is the coupling between a vector filed and a -form and denotes the interior product. The bundle inherits the non-degenerate symmetric bilinear form with signature defined by

 ⟨V+ξ,W+η⟩=12(η(V)+ξ(W)).
###### Definition 2.1.

An -twisted generalized complex structure on is an endmorphism of of the bundle satisfying the following properties:

1. ;

2. preserves the symmetric bilinear form ;

3. the subbundle of the complexified bundle which consists of -eigenvectors of is closed under the Courant bracket .

We call a generalized complex structure on if a -form vanishes at every point in .

The symmetric bilinear form yields the Clifford algebra of the complexified bundle . The action of the Clifford algebra of on the space of complex differential forms is given by

 (V+ξ)⋅ρ=iVρ+ξ∧ρ,

where denotes the interior product of by a vector filed and is the wedge product of by a -form . An -twisted generalized complex structure yields the canonical line bundle which consists of differential forms annihilated by elements of . The canonical line bundle satisfies the following conditions:

1. For any point , is written as , where are real -forms and are linearly independent complex -forms.

2. (non-degeneracy condition) the top degree part is not equal to for any , where denotes the complex conjugate of and is the involution given by reversal of order of forms, i.e., .

3. (integrability condition) a section of satisfies the following:

 d^ρ+H∧^ρ=(V+ξ)⋅^ρ,

where is a section of .

###### Definition 2.2.

The lowest degree of a differential form in at a point is called the type number of at .

Conversely, a line bundle satisfying the conditions (a), (b) and (c) gives an -twisted generalized complex structure by the following: It follows from the condition (a) that the subbundle which consists of annihilators of is maximal isotropic with respect to . The condition (b) is equivalent to the condition which gives rise to a direct sum decomposition . We define an endmorphism on by

 J(φ)={√−1φ(if φ∈L),−√−1φ(if φ∈¯¯¯¯L).

It follows that is compatible with the bilinear form . The condition (c) implies that is closed under the Courant bracket . Thus is an -twisted generalized complex structure on .

###### Remark 2.3.

If a -manifold admits an -twisted generalized complex structure , then admits an -twisted generalized complex structure for any closed -form which is cohomologous to . Indeed, there exists a -form such that is equal to and then is a line bundle which satisfies the conditions (a) and (b). If satisfies , then it follows that

 d(exp(−θ)⋅^ρ)+H′∧(exp(−θ)⋅^ρ)=(V′+ξ′)⋅(exp(−θ)⋅^ρ),

where Thus the line bundle satisfies the condition (c) for . In particular, if the cohomology group is trivial, then any twisted generalized complex structure can be changed into the untwisted one.

Let be a generalized complex -manifold with even type numbers. The natural projection gives rise to the canonical section . The type number of jumps from to when one goes into zeros of the section . We call the set of zeros the type changing loci of . A point is said to be nondegenerate if is a nondegenerate zero of . Cavalcanti and Gualtieri proved in [Cavalcanti_Gualtieri_2006] that any compact component of the type changing loci of which consists of nondegenerate points is a smooth elliptic curve whose complex structure is induced by .

## 3. C∞-logarithmic transformations for generalized complex 4-manifolds

Let be an embedded torus in a -manifold with trivial normal bundle. We denote by a regular neighborhood of , which is diffeomorphic to . We remove from and glue by a diffeomorphism . The procedure as above is called a -logarithmic transformation on along which yields a manifold . As explained in [Gompf_Stipsicz], the circle determines the diffeomorphism type of . A logarithmic transformation is said to be trivial if the circle is null-homotopic in . Note that a trivial logarithmic transformation on along does not change the diffeomorphism type of , that is, is diffeomorphic to .

###### Theorem 3.1.

Let be a twisted generalized complex structure on a -manifold and an embedded in a symplectic torus with trivial normal bundle. For any diffeomorphism , the manifold obtained from by a logarithmic transformation on admits a twisted generalized complex structure which coincides with on the complement . Moreover, if the logarithmic transformation is not trivial, has a non-empty type changing locus in and .

Since is diffeomorphic to , is obtained by an orientation preserving self-diffeomorphism of . Any matrix SL gives an orientation preserving self-diffeomorphism of as follows:

 ∂D2×T2∋(θ1,θ2,θ3)↦(θ1,θ2,θ3)A∈∂D2×T2,

where we identify the manifold with and denotes coordinates of with and . It is known that any orientation preserving self-diffeomorphism of is isotopic to the diffeomorphism represented by a matrix in SL.

###### Proof of Theorem 3.1.

If the logarithmic transformation determined by is trivial, the statement is obvious. We assume that the logarithmic transformation is not trivial.

Let be a symplectic form of which induces the generalized complex structure and is symplectic with respect to . By Weinstein’s neighborhood theorem [Weinstein], we can take a symplectomorphism:

 Θ:(νT,ωT)→(D2×T2,σC),

where denotes the unit disk and is the quotient with a coordinate and for the constant . Using this identification, the attaching map can be regarded as a matrix SL. Any matrix SL induces a self-diffeomorphism . Since the map preserves the form and the diffeomorphism type of is determined by the first row of , we can assume that is equal to the following matrix:

 ⎛⎜⎝m0p010a0b⎞⎟⎠

(see [Gompf_Stipsicz]), where are integers which satisfy . The first row of is and it follows that since is not trivial. Thus we can take and satisfying the condition

 (1) mbs−pa≠0\rm for all s∈[0,1]

by replacing by for a suitable integer if necessary. Let be the annulus for . We define a diffeomorphism as follows:

 Ψ(r,θ1,θ2,θ3)=(√log(er),mθ1+aθ3,θ2,pθ1+bθ3),

where and . The manifold is diffeomorphic to the following manifold:

 XΨ=(X∖Int(νT))∪ΨD2×T2.

Thus it suffices to construct a twisted generalized complex structure on which satisfies the conditions in Theorem 3.1.

We denote by the following form:

 z1exp(−mC2ϱ(|z1|2)dz1z1∧d¯z1¯z1−bCdz2∧d¯z2+C(a2−p)dz1z1∧dz2−C(a2+p)dz1z1∧d¯z2),

where is a monotonic increasing function which satisfies if and if . The form satisfies the condition (a) in Section 2. It follows form the condition (1) that the top-degree part is not trivial. Since is -closed on , the form satisfies the condition (c) in Section 2. Thus gives a generalized complex structure on . Denote by and the real part and the imaginary part of the degree- part of the form , respectively. Then it follows from a direct calculation that the pullback is equal to . We take a monotonic decreasing function which satisfies for and for , where is a sufficiently small number. We define a -form by

 ~B={Ψ−1∗(~ϱ(|z1|2)B)%onνT∖T,0on X∖νT.

Then the manifold admits a twisted generalized complex structure such that is a local section of the canonical bundle . Since gives a generalized complex structure on , the form induces a -twisted generalized complex structure. Since the -form is equal to , we obtain a twisted generalized complex structure on which satisifies the conditions in Theorem 3.1. This completes the proof of Theorem 3.1. ∎

## 4. Numbers of components of type changing loci

Let be a generalized complex -manifold and a symplectic torus with respect to . According to Theorem 3.1, the -manifold obtained from by a logarithmic transformation on admits a twisted generalized complex structure. Moreover, if the logarithmic transformation is not trivial, the number of components of type changing loci is increased by . Using this observation, we will prove in this section that several -manifolds admit twisted generalized complex structures with arbitrarily many components of type changing loci.

### 4.1. Elliptic surfaces

Let be a genus- Lefschetz fibration over with a Lefschetz singularity whose vanishing cycle is non-separating. The total space of is called a fishtail neighborhood. The manifold contains a torus as a regular fiber of the fibration . We denote by a circle which bounds a section of the fibration . For a non-zero integer , we take an orientation preserving diffeomorphism of which maps the homology class to . Let be a manifold obtained by removing from and gluing it back using . Gompf [Gompf_2010] proved that there exists an orientation and fiber preserving diffeomorphism from to whose restriction on the boundary is the identity map for every . Every elliptic surface with non-zero Euler characteristic contains a fishtail neighborhood as a submanifold (see [Gompf_Stipsicz, Theorem 8.3.12], for example). Furthermore, a regular fiber in an elliptic surface is symplectic with respect to the symplectic structure of the elliptic surface. We eventually obtain:

###### Theorem 4.1.

For any non-negative integer , every elliptic surface with non-zero Euler characteristic admits a generalized complex structure with components of type changing loci.

###### Proposition 4.2.

For every and , the connected sum admits generalized complex structures with .

###### Proof.

Cavalcanti and Gualtieri constructed a generalized complex structure on the connected sum which is obtained from the Lefschetz fibration by a multiplicity logarithmic transformation on a regular fiber of the fibration. They further constructed a generalized complex structure on the manifold by blowing down (see [Cabalcanti_Gualtieri_2009, Example 5.3] for details). The Lefschetz fibration contains a fishtail neighborhood. It is easy to verify that we can make the support of all the above surgeries away from the fishtail neighborhood. Thus, we can apply multiplicity logarithmic transformations on regular fibers of the fishtail neighborhood arbitrarily many times so that the diffeomorphism type of is unchanged. In particular, we can construct generalized complex structures of with . Using , we can construct a generalized complex structure of the manifold with components of type changing loci for arbitrarily large (see [Cabalcanti_Gualtieri_2009, Theorem 3.3]). This completes the proof of Proposition 4.2. ∎

### 4.2. Multiplicity 0 logarithmic transformations

We take a small ball and denote by the manifold . Let be a Morse function satisfying the following properties:

1. ;

2. ;

3. has a unique critical point in whose index is equal to .

We denote by the -manifold . Define a map as follows:

 f(x,t)=(h(x),t)((x,t)∈M×I).

The set of critical points of is and that every critical point of is an indefinite fold. A regular fiber of in the higher-genus side of the folds is a torus.

###### Lemma 4.3.

Let be a -manifold which contains an embedded torus with self-intersection . Then, the manifold obtained from by a multiplicity logarithmic transformation on contains as a submanifold. Furthermore, if admits a twisted generalized complex structure which makes symplectic, then a regular fiber of is also symplectic with respect to the induced twisted generalized complex structure on .

###### Proof.

We take an identification . Take a small regular neighborhood of . Denote by the manifold obtained from by a multiplicity logarithmic transformation on , that is, . We can take a handle decomposition of so that is decomposed into a -handle , two -handles and a -handle . Using the method in [Gompf_Stipsicz, Subsection 6.2], we can draw a Kirby diagram of as described in the left side of Figure 1. According to [Baykur_2009, Section 2], a Kirby diagram of is as in the right side of Figure 1.

By these diagrams, it is clear that the manifold contains as a submanifold. Furthermore, these diagrams also show that the torus is contained in as a symplectic torus. This completes the proof of Lemma 4.3. ∎

Let be a vanishing cycle of the folds of . We denote by a circle which bounds a section of a fibration .

###### Lemma 4.4.

Let be an orientation preserving self-diffeomorphism of such that the induced map on the first homology group maps the homology class to the class , where is an integer. We denote by the manifold . Then, there exists an orientation preserving diffeomorphism satisfying the following properties:

1. the restriction is the identity map (note that this condition makes sense since is equal to as a set);

2. the following diagram commutes:

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