On K\"ahler-Einstein Currents

TL;DR

This paper demonstrates that a broad class of singular K"ahler metrics with Ricci curvature bounded below are indeed K"ahler currents, extending to singular K"ahler-Einstein metrics and K"ahler-Ricci solitons, and relates these to RCD spaces.

Contribution

It establishes that certain singular K"ahler metrics with Ricci bounds are K"ahler currents and connects approximability by smooth metrics to RCD space structures.

Findings

Singular K"ahler-Einstein metrics define K"ahler currents.

Results apply to singular K"ahler-Einstein metrics on klt pairs.

Approximation conditions imply the metric defines an RCD space.

Abstract

We show that a general class of singular K\"ahler metrics with Ricci curvature bounded below define K\"ahler currents. In particular the result applies to singular K\"ahler-Einstein metrics on klt pairs, and an analogous result holds for K\"ahler-Ricci solitons. In addition we show that if a singular K\"ahler-Einstein metric can be approximated by smooth metrics on a resolution whose Ricci curvature has negative part that is bounded uniformly in $L^p$ for $p > \frac{2n-1}{n}$, then the metric defines an RCD space.