SNC K\"ahler-Einstein metrics and RCD spaces

TL;DR

This paper demonstrates that K"ahler-Einstein metrics with cone singularities along SNC divisors can define RCD spaces, providing new examples of Einstein RCD spaces, including non-compact manifolds with specific tangent cone structures.

Contribution

It establishes the connection between SNC K"ahler-Einstein metrics and RCD spaces, including the existence of non-compact examples with prescribed tangent cone links.

Findings

Existence of smooth non-compact 4-manifolds with ALE Ricci-flat RCD(0,4) metrics.

Construction of Einstein RCD spaces from SNC K"ahler-Einstein metrics.

Answering D. Semola's question on tangent cone structures.

Abstract

We show that K\"ahler-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 $S^3/\Gamma$ 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.