Existence and deformations of Kahler-Einstein metrics on smoothable Q-Fano varieties

TL;DR

This paper establishes the existence of Kahler-Einstein metrics on certain smoothable Q-Fano varieties and analyzes their behavior during smoothings using Gromov-Hausdorff convergence.

Contribution

It proves the existence of Kahler-Einstein metrics on Q-Gorenstein smoothable, K-polystable Q-Fano varieties and describes their deformation behavior.

Findings

Existence of Kahler-Einstein metrics on Q-Gorenstein smoothable Q-Fano varieties.

Description of metric behavior under Q-Gorenstein smoothings.

Metrics converge in the Gromov-Hausdorff sense during smoothings.

Abstract

We prove the existence of Kahler-Einstein metrics on Q-Gorenstein smoothable, K-polystable Q-Fano varieties, and we show how these metrics behave, in the Gromov-Hausdorff sense, under Q-Gorenstein smoothings.