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Hilbert schemes of K3 surfaces are dense in moduli

Eyal Markman, Sukhendu Mehrotra

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.

Hilbert schemes of K3 surfaces are dense in moduli

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.

Paper Structure

This paper contains 4 sections, 14 theorems, 18 equations.

Table of Contents

  1. Introduction
  2. Density of periods
  3. Density in moduli
  4. Density of generalized Kummer varieties

Key Result

Theorem 1.1

The locus in ${\mathfrak M}_\Lambda^0$, consisting of marked pairs $(X,\eta)$, where $X$ is isomorphic to the Hilbert scheme $S^{[n]}$, for some projective $K3$ surface $S$, is dense in ${\mathfrak M}_\Lambda^0$.

Theorems & Definitions (29)

  • Theorem 1.1
  • Lemma 2.1
  • Lemma 2.2
  • Remark 2.3
  • Lemma 2.4
  • proof
  • proof
  • Definition 3.1
  • Theorem 3.2
  • Lemma 3.3
  • ...and 19 more