Remarks on Singular K\"ahler-Einstein Metrics

TL;DR

This paper investigates two notions of singular K"ahler-Einstein metrics on complex varieties, establishing equivalences and properties of the underlying singularities, especially in the context of Ricci flat cone metrics and RCD spaces.

Contribution

It proves the equivalence of two notions of singular K"ahler-Einstein metrics in certain settings and characterizes the singularities as log terminal, extending to RCD spaces.

Findings

Weaker and stronger notions of singular K"ahler-Einstein metrics are equivalent in Ricci flat cone cases.

The underlying variety has log terminal singularities under these conditions.

The method applies to general singular K"ahler-Einstein spaces that are RCD spaces.

Abstract

We study two different natural notions of singular K\"ahler-Einstein metrics on normal complex varieties. In the setting of singular Ricci flat K\"ahler cone metrics that arise as non-collapsed limits of sequences of K\"ahler-Einstein metrics or K\"ahler-Ricci flows, we show that an a priori weaker notion is equivalent to the stronger one introduced by Eyssidieux-Guedj-Zeriahi, and in particular the underlying variety has log terminal singularities in this case. Our method applies to more general singular K\"ahler-Einstein spaces as well, assuming that they define RCD spaces.