On positivity of CM line bundles on the moduli space of klt good minimal models with $\kappa=1$

Abstract

We study the positivity of CM line bundles on the coarse moduli space of Kawamata log terminal (klt) good minimal models with Kodaira dimension one. We prove that the seminormalization of the moduli space is quasi-projective under a mild assumption on the general fibers of good minimal models. Moreover, we show that the CM line bundle becomes ample after normalization. A key new ingredient is the construction of a moduli space of numerical equivalence classes, which is an extension of the work of Viehweg and allows us to bypass the failure of quasi-finiteness in the approach of the previous work by Hashizume and the author. We also establish the projectivity of the moduli space of $\epsilon$-stable quotients, which is introduced by Toda, to a projective space, which plays a central role in our method. This particular situation is encompassed by our general framework of K-moduli of quasimaps.