Remarks on Singular Kähler-Einstein Metrics

TL;DR

This work analyzes singular Kähler-Einstein metrics on normal varieties by comparing the Eyssidieux-Guedj-Zeriahi (EGZ) notion with a weaker approach and proving equivalence under log terminal singularities. It introduces the rough Kähler-Einstein setting with an $RCD(bla,n)$ metric completion and develops elliptic estimates for $L^m$-sections, then constructs peaked holomorphic sections that extend to line bundles, yielding $$-Cartier and log-terminal conclusions. The results apply to noncollapsed limits of Kähler-Einstein manifolds and Kähler-Ricci flows, showing tangent cones are Ricci-flat Kähler cones with algebraic volume ratios and resolving related conjectures on limits and tangent flows. Overall, the paper links geometric-analytic regularity on singular spaces with algebraic-geometric consequences, enabling precise singular KE theory in non-smooth settings.

Abstract

We study two different natural notions of singular Kähler-Einstein metrics on normal complex varieties. In the setting of singular Ricci flat Kähler cone metrics that arise as non-collapsed limits of sequences of Kähler-Einstein metrics or Kähler-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 Kähler-Einstein spaces as well, assuming that they define RCD spaces.

Theorems & Definitions (30)

• Definition 1.2
• Theorem 1.3
• Remark 1.4
• Theorem 1.5
• Theorem 1.6
• Remark 1.7
• Lemma 2.1
• proof
• Lemma 2.2
• proof