Geometry and Topology of the space of K\"ahler metrics on singular varieties

TL;DR

This paper investigates the geometric structure of the space of K"ahler metrics on singular varieties, extending known results to normal spaces and applying the findings to characterize K"ahler-Einstein metrics on Fano varieties.

Contribution

It generalizes the metric properties of K"ahler metric spaces to singular varieties and provides an analytic characterization of K"ahler-Einstein metrics on ano varieties.

Findings

Extended Calabi, Chen, Darvas results to singular spaces

Characterized existence of K"ahler-Einstein metrics on ano varieties

Applied concepts to toric varieties

Abstract

Let $Y$ be a compact K\"ahler normal space and $\alpha \in H^{1,1}(Y,\mathbb{R})$ a K\"ahler class. We study metric properties of the space $\mathcal{H}_\alpha$ of K\"ahler metrics in $\alpha$ using Mabuchi geodesics. We extend several results by Calabi, Chen, Darvas previously established when the underlying space is smooth. As an application we analytically characterize the existence of K\"ahler-Einstein metrics on $\mathbb{Q}$-Fano varieties, generalizing a result of Tian, and illustrate these concepts in the case of toric varieties.