K-moduli of quasimaps and quasi-projectivity of moduli of K-stable Calabi-Yau fibrations over curves

Abstract: We construct a projective K-moduli space of quasimaps with a certain log Fano condition. Moreover, we investigate relationships between the K-moduli of quasimaps and the K-moduli of Calabi-Yau fibrations over curves of negative Kodaira dimension constructed by the authors [HaHa23] when general fibers are Abelian varieties or irreducible holomorphic symplectic manifolds. We show that there is a quasi-finite morphism from the K-moduli of Calabi-Yau fibrations to the K-moduli of quasimaps and the CM line bundle of the K-moduli of Calabi-Yau fibrations converges to the CM line bundle of the K-moduli of quasimaps. As a corollary, we obtain the entire quasi-projectivity of K-moduli of Calabi-Yau fibrations in this case.