On the Kodaira dimension of the moduli of deformation generalised Kummer varieties

We prove general-type results for orthogonal modular varieties associated with moduli spaces of compact hyperkhler manifolds of deformation generalised Kummer type ('deformation generalised Kummer varieties' ).More precisely, 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 certain bounds.As a corollary, we conclude that the corresponding moduli spaces are also of general type.