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

Abstract

In this paper, we derive from deep results due to Clozel and Ullmo a sharp density result 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 the Mori cone of curves and of the associated extremal birational contractions, and a refinement of Hassett’s result on cubic fourfolds whose Fano variety of lines is isomorphic to a Hilbert scheme of two 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.