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Kodaira dimension of moduli spaces of hyperk\"ahler varieties

Ignacio BarrosPietro BeriEmma BrakkeeLaure Flapan
Dec 2022
摘要
We study the Kodaira dimension of moduli spaces of polarized hyperk\"ahlervarieties deformation equivalent to the Hilbert scheme of points on a K3surface or 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.
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