Period map for non-compact holomorphically symplectic manifolds

TL;DR

This paper investigates the deformation theory of non-compact holomorphic symplectic manifolds, establishing conditions for the existence and smoothness of their deformation spaces and describing their structure in terms of second cohomology.

Contribution

It extends deformation theory to non-compact holomorphic symplectic manifolds, showing the existence, smoothness, and explicit description of their deformation spaces under mild conditions.

Findings

Deformation space exists and is smooth for certain non-compact manifolds.

Deformation space embeds into second cohomology group.

Explicit description of moduli space as a formal power series ring.

Abstract

We study the deformations of a holomorphic symplectic manifold $M$, not necessarily compact, over a formal ring. We show (under some additional, but mild, assumptions on $M$) that the coarse deformation space exists and is smooth, finite-dimensional and naturally embedded into $H^2(M)$. For a holomorphic symplectic manifold $M$ which satisfies $H^1(M,{\cal O}_M) = H^2(M,{\cal O}_M)=0$, the coarse moduli of formal deformations is isomorphic to $\Spec\C[[t_1, ..., t_n]]$, where $t_1$, ... $t_n$ are coordinates in $H^2(M)$. This revised version contains one minor improvement: exposition in Subsection 5.1 has been made more detailed and rigourous.