General type results for moduli of deformation generalised Kummer varieties

TL;DR

This paper extends previous results by proving that orthogonal modular varieties associated with moduli spaces of deformation generalized Kummer hyperk"ahler manifolds are of general type without requiring the polarisation degree to be square-free.

Contribution

It removes the square-free restriction on the degree of polarisation for orthogonal modular varieties related to deformation generalized Kummer varieties.

Findings

Orthogonal modular varieties are of general type for all degrees 2d.

The square-free condition on 2d is no longer necessary.

General type results now apply to a broader class of moduli spaces.

Abstract

In arXiv:1710.01672, we obtained general type results for orthogonal modular varieties associated with moduli spaces of compact hyperk\"ahler manifolds of deformation generalised Kummer type (also known as 'deformation generalised Kummer varieties'). The orthogonal modular varieties were defined in terms of an integer 2d, corresponding to the degree of polarisation of the associated hyperk\"ahler manifolds. In arXiv:1710.01672, we showed these orthogonal modular varieties are of general type when 2d is square-free and sufficiently large. Here we show that the square-free condition can be removed.