On boundedness and moduli spaces of K-stable Calabi-Yau fibrations over curves

TL;DR

This paper proves boundedness of polarized Calabi-Yau fibrations over curves with fixed fiber volumes, constructs their moduli space, and demonstrates uniform K-stability and existence of cscK metrics for members.

Contribution

It establishes boundedness results and constructs a moduli space for K-stable Calabi-Yau fibrations over curves, with applications to stability and metric properties.

Findings

Boundedness of polarized Calabi-Yau fibrations with fixed volumes.

Construction of a separated coarse moduli space for these fibrations.

Uniform K-stability and existence of cscK metrics for members of the moduli.

Abstract

We show boundedness of polarized Calabi--Yau fibrations over curves only with fixed volumes of general fibers and Iitaka volumes. As its application, we construct a separated coarse moduli space of K-stable Calabi-Yau fibrations over curves in an adiabatic sense [Hat22b] and show that all members (resp. smooth members) of the moduli are simultaneously uniformly K-stable (resp. have cscK metrics) for a certain choice of polarizations.