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SNC Kähler-Einstein metrics and RCD spaces

Martin de Borbon, Cristiano Spotti

TL;DR

The paper proves that Kähler–Einstein metrics with cone singularities along SNC divisors define $RCD(\lambda,2n)$ spaces in both compact and certain non-compact (ALE) settings. The authors leverage Honda and Honda–Sun’s almost-smooth characterizations, establishing an almost-smooth metric measure structure, Sobolev–Lipschitz and $L^2$–strong compactness, and Schauder regularity for conical metrics, to derive the $RCD$ property. In the compact case, this yields a broad class of Einstein $RCD$ spaces, while in the non-compact ALE case they construct Ricci-flat $RCD(0,4)$ spaces with prescribed links $S^3/\Gamma$, answering a question of Semola. These results extend Cheeger–Colding limit theory into the Kähler setting and provide a robust framework for studying moduli, limits, and tangent-cone structure of Einstein metrics with SNC singularities.

Abstract

We show that Kähler-Einstein metrics with cone singularities along simple normal crossing (SNC) divisors define RCD spaces, both in the compact setting and in certain non-compact cases, thereby producing many examples of Einstein RCD spaces. In particular, we show the existence of smooth non-compact $4$-manifolds carrying ALE Ricci-flat RCD$(0,4)$ metrics with any space form $S^3/Γ$ as the link of the tangent cone at infinity, answering a question raised by D. Semola. Our proofs rely on the characterization of RCD spaces in the almost-smooth setting due to S. Honda and Honda-Sun.

SNC Kähler-Einstein metrics and RCD spaces

TL;DR

The paper proves that Kähler–Einstein metrics with cone singularities along SNC divisors define spaces in both compact and certain non-compact (ALE) settings. The authors leverage Honda and Honda–Sun’s almost-smooth characterizations, establishing an almost-smooth metric measure structure, Sobolev–Lipschitz and –strong compactness, and Schauder regularity for conical metrics, to derive the property. In the compact case, this yields a broad class of Einstein spaces, while in the non-compact ALE case they construct Ricci-flat spaces with prescribed links , answering a question of Semola. These results extend Cheeger–Colding limit theory into the Kähler setting and provide a robust framework for studying moduli, limits, and tangent-cone structure of Einstein metrics with SNC singularities.

Abstract

We show that Kähler-Einstein metrics with cone singularities along simple normal crossing (SNC) divisors define RCD spaces, both in the compact setting and in certain non-compact cases, thereby producing many examples of Einstein RCD spaces. In particular, we show the existence of smooth non-compact -manifolds carrying ALE Ricci-flat RCD metrics with any space form as the link of the tangent cone at infinity, answering a question raised by D. Semola. Our proofs rely on the characterization of RCD spaces in the almost-smooth setting due to S. Honda and Honda-Sun.
Paper Structure (11 sections, 13 theorems, 22 equations)

This paper contains 11 sections, 13 theorems, 22 equations.

Table of Contents

  1. Introduction
  2. Compact case
  3. Conical Kähler metrics: basic definitions
  4. Almost smooth metric measure space structure
  5. Sobolev to Lipschitz and $L^2$-strong compactness
  6. Schauder estimates
  7. Proof of Theorem \ref{['KEcpt']}
  8. Non-compact case
  9. ALE conical Kähler metrics
  10. Volume doubling and Poincaré inequality
  11. Proof of Theorem \ref{['ALE4d']}

Key Result

Theorem 1.1

A conical Kähler--Einstein metric on a compact simple normal crossing pair $(X^n,\sum_i(1-\beta_i)D_i)$ with $\beta_i \in (0,1)$ defines an $RCD(\lambda, 2n)$ space.

Theorems & Definitions (33)

  • Theorem 1.1
  • Remark
  • Theorem 1.2
  • Definition 2.1
  • Lemma 2.2
  • proof
  • Lemma 2.3
  • proof
  • Lemma 2.4
  • proof
  • ...and 23 more