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

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 $\mathrm{O}(2,n)$. As a corollary, we conclude that the corresponding moduli spaces are also of general type.