Kodaira dimension of moduli spaces of hyperk\"ahler varieties

TL;DR

This paper investigates the Kodaira dimension of moduli spaces of polarized hyperk"ahler varieties, extending previous results to higher dimensions and divisibility, and establishing conditions under which these moduli spaces are of general type.

Contribution

It generalizes earlier work on the Kodaira dimension of hyperk"ahler moduli spaces to higher dimensions and divisibility cases, providing new lower bounds for their general type.

Findings

Established lower bounds on degrees for moduli spaces to be of general type.

Extended results to higher-dimensional hyperk"ahler varieties.

Provided conditions for the moduli spaces to have maximal Kodaira dimension.

Abstract

We study the Kodaira dimension of moduli spaces of polarized hyperk\"ahler varieties deformation equivalent to the Hilbert scheme of points on a K3 surface or to O'Grady's ten dimensional variety. This question was studied by Gritsenko-Hulek-Sankaran in the cases of $K3^{[2]}$ and OG10 type when the divisibility of the polarization is one. We generalize their results to higher dimension and divisibility. As a main result, for almost all dimensions $2n$ we provide a lower bound on the degree such that for all higher degrees, every component of the moduli space of polarized hyperk\"ahler varieties of $K3^{[n]}$ type is of general type.