Singular symplectic surfaces

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 $\ddot{I}ƒ$ on the smooth locus, and for which every finite quasi-étale covering has the algebra of reflexive forms spanned by the reflexive pull-back of $\ddot{I}ƒ$. 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.

52 pages; v2:Section 7 has been revised, Added an Appendix, minor other revisions. To appear in BZAG