General type results for moduli of deformation generalised Kummer varieties

In Dawes [Algebr. Geom. 12(2025), no. 3, 601–660], families of orthogonal modular varieties associated with moduli spaces of compact hyperkähler manifolds of deformation generalized Kummer type (also known as “deformation generalized Kummer varieties” ) were studied. The orthogonal modular varieties were defined for an even integer , corresponding to the degree of polarization of the associated hyperkähler manifolds. It was shown in Dawes [Algebr. Geom. 12(2025), no. 3, 601–660] that the modular varieties are of general type when is square‐free and sufficiently large. The purpose of this paper is to show that the square‐free condition can be removed.