Positivity of the CM line bundle for K-stable log Fanos

TL;DR

This paper proves the positivity (bigness) of the CM line bundle for families of log Fano varieties with stable fibers, extending previous results to the logarithmic case.

Contribution

It generalizes the bigness of the CM line bundle to the setting of log Fano varieties with maximal variation and uniformly K-stable fibers.

Findings

Proves the bigness of the Chow-Mumford line bundle in the log Fano setting.

Extends Codogni and Patakfalvi's theorem to logarithmic varieties.

Demonstrates the positivity for families with maximal variation and K-stable fibers.

Abstract

We prove the bigness of the Chow-Mumford line bundle associated to a $\mathbb{Q}$-Gorenstein family of log Fano varieties of maximal variation with uniformly K-stable general geometric fibers. This result generalizes a recent theorem of Codogni and Patakfalvi to the logarithmic setting.