Hilbert schemes of K3 surfaces are dense in moduli

TL;DR

This paper proves that the moduli space of Hilbert schemes of points on K3 surfaces is dense within the broader moduli space of irreducible holomorphic symplectic manifolds, extending to generalized Kummer manifolds.

Contribution

It establishes the density of Hilbert schemes of points on K3 surfaces in their moduli space, including analogous results for generalized Kummer manifolds.

Findings

Hilbert schemes of n points on K3 surfaces are dense in their moduli space

Density result also proven for generalized Kummer manifolds

Advances understanding of the structure of moduli spaces for holomorphic symplectic manifolds

Abstract

We prove that the locus of Hilbert schemes of n points on a projective K3 surface is dense in the moduli space of irreducible holomorphic symplectic manifolds of that deformation type. The analogous result for generalized Kummer manifolds is proven as well.