Holomorphic curves in log-symplectic manifolds

TL;DR

This paper develops the theory of holomorphic curves in log-symplectic manifolds, constructing moduli spaces, classifying certain 4-manifolds, and applying symplectic field theory tools to understand their structure.

Contribution

It introduces the construction of moduli spaces of holomorphic curves in log-symplectic manifolds and classifies symplectically ruled log-symplectic 4-manifolds.

Findings

Classification of symplectically ruled log-symplectic 4-manifolds

Obstructions to contact boundary components

Application of symplectic field theory tools

Abstract

Log-symplectic structures are Poisson structures that are determined by a symplectic form with logarithmic singularities. We construct moduli spaces of curves with values in a log-symplectic manifold. Among the applications, we classify symplectically ruled log-symplectic $4$ manifolds (both orientable and non-orientable), and obstruct the existence of contact boundary components, in analogy with well-known theorems by McDuff. Moreover, we study certain log-symplectically ruled surfaces, using tools from symplectic field theory.