On positivity of the CM line bundle on K-moduli spaces

TL;DR

This paper proves the ampleness of the CM line bundle on certain K-moduli spaces of Fano varieties, establishing projectivity of the moduli space and introducing a new invariant for K-stability testing.

Contribution

It demonstrates the positivity of the CM line bundle on a broad class of K-moduli spaces and develops a new invariant for analyzing K-stability of Fano varieties.

Findings

CM line bundle is ample on proper subspaces of K-moduli

Moduli space of smoothable K-polystable Fano varieties is projective

Introduces a new invariant for K-stability testing

Abstract

In this paper, we consider the CM line bundle on the K-moduli space, i.e., the moduli space parametrizing K-polystable Fano varieties. We prove it is ample on any proper subspace parametrizing reduced uniformly K-stable Fano varieties which conjecturally should be the entire moduli space. As a corollary, we prove that the moduli space parametrizing smoothable K-polystable Fano varieties is projective. During the course of proof, we develop a new invariant for filtrations which can be used to test various K-stability notions of Fano varieties.