Special K-stability and positivity of CM line bundles

TL;DR

This paper proves the ampleness of the CM line bundle for certain K-stable families and applies this to establish the projectivity of specific moduli subspaces of K-stable fibrations.

Contribution

It demonstrates the ampleness of the CM line bundle in the context of special K-stability and derives projectivity results for moduli spaces of K-stable fibrations.

Findings

CM line bundle is ample for K-stable families with maximal variation

Proper subspaces of moduli spaces of K-stable fibrations are projective

Provides new tools for studying moduli of K-stable varieties

Abstract

We show that the CM line bundle on a proper family parametrizing specially K-stable varieties with maximal variation is ample. As an application, we show projectivity of any proper subspace of the coarse moduli space of uniformly adiabatically K-stable klt--trivial fibrations over curves constructed in [HH23].