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We study the Kodaira dimension of moduli spaces of polarized hyperk\"ahlervarieties deformation equivalent to the Hilbert scheme of points on a K3surface or to O'Grady's ten dimensional variety. This question was studied byGritsenko-Hulek-Sankaran in the cases of $K3^{[2]}$ and OG10 type when thedivisibility of the polarization is one. We generalize their results to higherdimension and divisibility. As a main result, for almost all dimensions $2n$ weprovide a lower bound on the degree such that for all higher degrees, everycomponent of the moduli space of polarized hyperk\"ahler varieties of$K3^{[n]}$ type is of general type.

Kodaira dimension of moduli spaces of hyperk\"ahler varieties