Positivity of the CM line bundle for families of K-stable klt Fano varieties

TL;DR

This paper proves semi-positivity and positivity of the CM line bundle for families of K-stable klt Fano varieties, advancing understanding of their moduli spaces and applications to classification.

Contribution

It establishes semi-positivity and positivity results for the CM line bundle in general singular settings, including K-semistable and K-stable cases, using algebraic and probabilistic methods.

Findings

Proved semi-positivity of the CM line bundle for K-semistable families.

Proved positivity of the CM line bundle for uniformly K-stable families.

Applied results to the classification of Fano varieties.

Abstract

The Chow-Mumford (CM) line bundle is a functorial line bundle on the base of any family of klt Fano varieties. It is conjectured that it yields a polarization on the moduli space of K-poly-stable klt Fano varieties. Proving ampleness of the CM line bundle boils down to showing semi-positivity/positivity statements about the CM-line bundle for families with K-semi-stable/K-polystable fibers. We prove the necessary semi-positivity statements in the K-semi-stable situation, and the necessary positivity statements in the uniform K-stable situation, including in both cases variants assuming K-stability only for general fibers. Our statements work in the most general singular situation (klt singularities), and the proofs are algebraic, except the computation of the limit of a sequence of real numbers via the central limit theorem of probability theory. We also present an application to the classification of Fano varieties. Additionally, our semi-positivity statements work in general for log-Fano pairs.