On the Kodaira Dimension of the Moduli of Deformation Generalised Kummer Varieties

TL;DR

This paper establishes that certain moduli spaces of deformation generalized Kummer fourfolds are of general type under specific conditions on the polarization degree, advancing understanding of their geometric complexity.

Contribution

It proves that moduli spaces of deformation generalized Kummer fourfolds are of general type when the polarization degree meets certain prime or square-free conditions, depending on the existence of low-weight cusp forms.

Findings

Moduli spaces are of general type for prime or square-free degrees.

Results depend on the existence of low-weight cusp forms.

Provides bounds on the degree for general type classification.

Abstract

We prove general type results for orthogonal modular varieties associated with the moduli of compact hyperk\"ahler manifolds of deformation generalised Kummer type ('deformation generalised Kummer varieties'). In particular, we consider moduli spaces of deformation generalised Kummer fourfolds with split-polarisation of degree $2d$. Our main result is that when $d$ is prime or $2d$ is square-free then the associated modular varieties are of general type when $d$ exceeds bounds we determine, subject to the existence of certain low-weight cusp forms for $\operatorname{O}(2,n)$. As a corollary, we conclude that the corresponding moduli spaces are also of general type.