Density of Noether-Lefschetz loci of polarized irreducible holomorphic symplectic varieties and applications

TL;DR

This paper proves the density of Noether-Lefschetz loci in the moduli space of polarized irreducible holomorphic symplectic varieties, with applications to rational curves, divisor cones, and cubic fourfolds.

Contribution

It establishes the density of Noether-Lefschetz loci in moduli spaces of IHS varieties using deep results, and explores several geometric applications.

Findings

Density of Hilbert schemes of points on K3 surfaces in moduli spaces

Density of generalized Kummer varieties in their moduli spaces

Refinement of Hassett's results on cubic fourfolds

Abstract

In this note we derive from deep results due to Clozel-Ullmo the density of Noether-Lefschetz loci inside the moduli space of marked (polarized) irreducible holomorphic symplectic (IHS) varieties. In particular we obtain the density of Hilbert schemes of points on projective $K3$ surfaces and of projective generalized Kummer varieties in their moduli spaces. We present applications to the existence of rational curves on projective deformations of such varieties, to the study of relevant cones of divisors, and a refinement of Hassett's result on cubic fourfolds whose Fano variety of lines is isomorphic to a Hilbert scheme of 2 points on a K3 surface. We also discuss Voisin's conjecture on the existence of coisotropic subvarieties on IHS varieties and relate it to a stronger statement on Noether-Lefschetz loci in their moduli spaces.