Moduli spaces of irreducible symplectic manifolds

TL;DR

This paper investigates the structure of moduli spaces of polarized irreducible symplectic manifolds, revealing conditions under which these spaces are of general type, especially for certain degrees related to K3 surface Hilbert schemes.

Contribution

It establishes a connection between moduli spaces of polarized symplectic manifolds and locally symmetric varieties, identifying when these moduli spaces are of general type.

Findings

Moduli space of degree 2 Hilbert schemes of K3 surfaces is of general type for degree d ≥ 12.

Comparison with orthogonal type symmetric varieties provides new insights into the geometry of these moduli spaces.

The work advances understanding of the classification of irreducible symplectic manifolds and their moduli.

Abstract

We study the moduli spaces of polarised irreducible symplectic manifolds. By a comparison with locally symmetric varieties of orthogonal type of dimension 20, we show that the moduli space of 2d polarised (split type) symplectic manifolds which are deformation equivalent to degree 2 Hilbert schemes of a K3 surface is of general type if d is at least 12.