Singular symplectic surfaces

TL;DR

This paper classifies singular irreducible symplectic surfaces, characterizes their properties, and proves that their Hilbert schemes of two points are irreducible symplectic varieties under certain conditions.

Contribution

It provides a complete classification of singular irreducible symplectic surfaces and establishes that their Hilbert schemes of two points are irreducible symplectic varieties in specific cases.

Findings

Classification of all singular irreducible symplectic surfaces.

Proof that the Hilbert scheme of two points on such surfaces is an irreducible symplectic variety.

Confirmation that the smooth locus being simply connected is a key condition.

Abstract

In this paper we classify all singular irreducible symplectic surfaces, i.e., compact, connected complex surfaces with canonical singularities that have a holomorphic symplectic form $\sigma$ on the smooth locus, and for which every finite quasi-\'etale covering has the algebra of reflexive forms spanned by the reflexive pull-back of $\sigma$. We moreover prove that the Hilbert scheme of two points on such a surface $X$ is an irreducible symplectic variety, at least in the case where the smooth locus of $X$ is simply connected.