Finiteness of polarized K3 surfaces and hyperk\"ahler manifolds

TL;DR

This paper proves the finiteness of polarized K3 surfaces and hyperk"ahler manifolds in moduli spaces without relying on the cone conjecture or Torelli theorem, using Baily-Borel type arguments.

Contribution

It provides a new proof of finiteness for polarized K3 and hyperk"ahler manifolds, independent of the cone conjecture and Torelli theorem, via moduli space geometry.

Findings

Finiteness of polarized K3 surfaces and hyperk"ahler manifolds established.

New proof does not depend on the cone conjecture or Torelli theorem.

Addresses finiteness in twistor families of CM type K3 surfaces.

Abstract

In the moduli space of polarized varieties the same unpolarized variety can occur multiple times However, for K3 surfaces, compact hyperk\"ahler manifolds, and abelian varieties the number is finite. This may be viewed as a consequence of the Kawamata-Morrison cone conjecture. In this note we provide a proof of this finiteness not relying on the cone conjecture and, in fact, not even on the global Torelli theorem. Instead, it uses the geometry of the moduli space of polarized varieties to conclude the finiteness by means of Baily-Borel type arguments. We also address related questions concerning finiteness in twistor families associated with polarized K3 surfaces of CM type.