On Kähler-Einstein Currents

TL;DR

This work answers when singular Kähler-Einstein metrics on klt pairs define Kähler currents and when the associated metric spaces satisfy synthetic Ricci lower bounds. It develops tame approximations with $L^1$ control of the Ricci curvature and harnesses heat-kernel techniques to pass to the limit, establishing the Kähler current property and lower bounds that yield current-level curvature control. It further extends the framework to singular Kähler-Ricci solitons and proves that, under a uniform $L^p$ bound on the negative Ricci part with $p>\frac{2n-1}{n}$, the metric completion of the regular locus is a non-collapsed $RCD(2n,\lambda)$ space, with implications for extremal metrics and symmetry groups. Overall, the paper strengthens the link between complex-analytic singularities, Monge-Ampère geometry, and synthetic Ricci curvature, providing robust tools for the metric geometry of singular KE spaces.

Abstract

We show that a general class of singular Kähler metrics with Ricci curvature bounded below define Kähler currents. In particular the result applies to singular Kähler-Einstein metrics on klt pairs, and an analogous result holds for Kähler-Ricci solitons. In addition we show that if a singular Kähler-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.

Theorems & Definitions (22)

• Theorem 1.1
• Theorem 1.2
• Theorem 1.3
• Definition 1.4
• Theorem 1.5
• Definition 2.1
• Theorem 2.2: See GPSS2
• Proposition 2.3
• Proposition 3.1
• proof