# Locally conformal symplectic blow-ups

###### Abstract.

In this paper, we study the blow-up of a locally conformal symplectic manifold. We show that there exists a locally conformal symplectic structure on the blow-up of a locally conformal symplectic manifold along a compact induced symplectic submanifold.

###### Key words and phrases:

Locally conformal symplectic manifolds, Locally conformal symplectic blow-ups###### 2010 Mathematics Subject Classification:

55B35, 55C55all

## 1. Introduction

Let be a smooth manifold. A symplectic form on is a -form satisfying: (1) and (2) is non-degenerate, i.e. for each the map

is an isomorphism. It is of importance to point out that the existence of the symplectic form on determines pieces of topological data: the de Rham cohomology of with even degrees are non-vanishing and the dimension of is even, denoted by , and there exists a homotopy class of reductions of the structural group of the tangent bundle to . In particular, if is a complex manifold and is the Kähler form of a Hermitian metric on then we say that is a Kähler manifold.

In a more general setting, a subclass of almost symplectic manifolds called locally conformal symplectic manifolds (LCS for short) was introduced and studied by Lee [Lee43], Liebermann[PL54] and Vaisman [IV76, IV85]. Intuitively, a locally conformal symplectic form is a non-degenerate 2-form which is conformally equivalent to a symplectic form locally. From a conformal point of view, locally conformal symplectic manifolds can be thought of the closest to symplectic manifolds. In particular, the locally conformal symplectic manifolds can serve as natural phase spaces of Hamiltonian dynamical systems and from the geometric aspect it appears in the study of contact manifolds and Jacobi manifolds (cf. [BK11, GL84, IV85]). Likewise, if is a complex manifold and the locally conformal symplectic form on is the Kähler form of a Hermitian metric then we say that is a locally conformal Kähler manifold (LCK for short)(cf. [DO98]). To make this more precisely, we have the following diagram explaining the relationships between symplectic/Kähler manifolds and locally conformal symplectic/Kähler manifolds:

It is well known that the blow-up is a very useful operation in symplectic/Kähler geometry.
In particular, the Kähler property is preserved under blow-ups.
In the symplectic category, it was McDuff [McDuff84] who first proved that the blow-up of a symplectic manifold along a compact symplectic submanifold also admits a symplectic structure,
moreover, using this symplectic blow-up technique she constructed the first simply-connected, symplectic manifold which is non-Kähler.
For locally conformal Kähler manifolds, Tricerri [FT82] and Vuletescu [VV09] proved that the blow-up of a locally conformal Kähler manifold at a point has a locally conformal Kähler structure.
In 2013, using the current theory on locally conformal Kähler manifolds, Ornea-Verbitsky-Vuletescu [OVV13] showed that the blow-up of a locally conformal Kähler manifold along a submanifold is locally conformal Kähler if and only if the submanifold is globally conformally equivalent to a Kähler submanifold.
In the locally conformal symplectic case, Y. Chen and the first named author [CY16] introduced the definition of locally conformal symplectic blow-up of points and proved that the locally conformal symplectic blow-ups of points also admit locally conformally symplectic structures.
Therefore, a natural problem is:
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What is the locally conformal symplectic blow-up along a submanifold?
*

The purpose of this paper is to study some birational properties of locally conformal symplectic manifolds. Inspired by the work of McDuff [McDuff84] we give the construction of the locally conformal symplectic blow-up. In addition, using the same methods of McDuff [McDuff84] and Ornea-Verbitsky-Vuletescu [OVV13] we prove the following result

###### Theorem 1.1.

Let be a locally conformal symplectic manifold and be a compact induced globally conformal symplectic submanifold of , and let along . Then also admits a locally conformal symplectic structure where . be the blow-up of

This paper is organized as follows. We devote Section 2 to the preliminary of locally conformal symplectic structures. In Section 3, we give the construction of locally conformal symplectic blow-up. This construction is based on the fact that the tangent bundle of a locally conformal symplectic manifold is a symplectic vector bundle. In Section 4, we give the proof of the main result (Theorem 1.1). Finally, we propose two further problems related to the locally conformal symplectic blow-up.

## 2. Locally conformal symplectic manifolds

In this section we give a rapid review of locally conformal symplectic manifolds. Assume that is a smooth manifold of dimension . Intuitively, a locally conformal symplectic structure on is a non-degenerate -form which is locally conformal to a symplectic form. More precisely, if there exists an open covering of and a family of smooth real-valued functions is a symplectic form on , i.e., then we say that is a locally conformal symplectic structure on . such that

Let then from definition we have

on . This implies that

(2.1) |

on . Likewise, consider the form we get

(2.2) |

on . Suppose that then from (2.1) and (2.2) we obtain

(2.3) |

on . Note that is non-degenerate and the wedge product with is injective on -forms, hence we obtain a globally defined closed -form on which satisfies

(2.4) |

Equivalently, we have

###### Definition 2.1 (Locally conformal symplectic structure).

Let be a smooth manifold of dimension . We say that a non-degenerate -form is a locally conformal symplectic structure, if there exists a closed -form such that

(2.5) |

The triple is called a locally conformal symplectic manifold.

Suppose that there exists another satisfying (2.5), then . From the Cartan lemma we get for some -form ; however, this leads to a contradiction with the non-degeneracy of . This implies that is uniquely determined by and we call it the Lee form of the LCS manifold. In particular, if is an exact -form, i.e. for some smooth function on then is called globally conformal symplectic (GCS for short) and it is straightforward to verify that is a symplectic form on .

###### Example 2.2.

Every LCK manifold is a LCS manifold. In particular, many well-known non-Kähler manifolds, such as the Hopf manifolds and the Inoue surfaces and so on, are LCK manifolds (cf. [DO98, Chapter 3]).

###### Example 2.3.

Let be a smooth manifold. Then the cotangent bundle is an open symplectic manifold with the symplectic form , where is the canonical -form on . If is a closed -form on , then is a LCS form on with the Lee form , where is an exact -form then is a GCS form. is the bundle map. Moreover, if

###### Example 2.4.

([BK11, Section 5]) Let be a compact contact manifold and let be a strict contactomorphism, then there exists a LCS structure on the mapping torus of with respect to . In particular, we can choose a 3-dimensional contact manifold such that admits no symplectic and complex structures. This gives rise to an example that is LCS and not LCK.

Let be the space of smooth forms on the LCS manifold .
We can define the Lichnerowicz differential
^{1}^{1}1In the case of LCK manifolds the differential is called the -twisted differential and the associated complex (cohomology) is called the Morse-Novikov complex (cohomology).
by

Furthermore, we have a complex

(2.6) |