Kodaira dimension of moduli spaces of hyperkähler varieties

We study the Kodaira dimension of moduli spaces of polarized hyperkähler varieties deformation equivalent to the Hilbert scheme of points on a K3 surface or to O’Grady’s ten dimensional variety. This question was studied by Gritsenko-Hulek-Sankaran in the cases of $K{3}^{[2]}$ and OG10 type when the divisibility of the polarization is one. We generalize their results to higher dimension and divisibility. As a main result, for almost all dimensions $2n$ we provide a lower bound on the degree such that for all higher degrees, every component of the moduli space of polarized hyperkähler varieties of $K{3}^{[n]}$ type is of general type.