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

We construct a projective K-moduli space of quasimaps with a certain log Fano condition, which is regarded as a rational map from ${\mathbb{P}}^1$ to a projective space. 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 when general fibers are Abelian varieties or irreducible holomorphic symplectic manifolds. As an application, we obtain the entire quasi-projectivity of the seminormalization and the ampleness of the CM line bundle on the normalization of the K-moduli space of Calabi-Yau fibrations in this case.