Examples and counter-examples of log-symplectic manifolds

TL;DR

This paper explores the topological characteristics of log-symplectic manifolds, providing examples, non-existence results, and surgical methods to construct and analyze such structures on various 4-manifolds.

Contribution

It introduces surgeries to generate log-symplectic manifolds from symplectic ones and characterizes when certain 4-manifolds admit these structures, highlighting their differences from symplectic manifolds.

Findings

Certain connected sums of CP^2 and its conjugate admit log-symplectic structures only if both parameters are positive.

Some symplectic manifolds do not admit log-symplectic structures, and vice versa.

Surgical techniques can produce log-symplectic structures on specific 4-manifolds.

Abstract

We study topological properties of log-symplectic structures and produce examples of compact manifolds with such structures. Notably we show that several symplectic manifolds do not admit log-symplectic structures and several log-symplectic manifolds do not admit symplectic structures, for example #m CP^2 # n bar(CP^2)$ has log-symplectic structures if and only if m,n>0 while they only have symplectic structures for m=1. We introduce surgeries that produce log-symplectic manifolds out of symplectic manifolds and show that for any simply connected 4-manifold M, the manifolds M # (S^2 \times S^2) and M # CP^2 # CP^2bar have log-symplectic structures and any compact oriented log-symplectic four-manifold can be transformed into a collection of symplectic manifolds by reversing these surgeries.