Degenerating conic K\"ahler-Einstein metrics to the normal cone

TL;DR

This paper studies the degeneration of conic K"ahler-Einstein metrics on Fano manifolds as the cone angle approaches a critical limit, revealing the emergence of Tian-Yau metrics in the limit.

Contribution

It proves the existence of conic K"ahler-Einstein metrics near the critical angle and identifies their limits, including Tian-Yau metrics, under certain automorphism conditions.

Findings

Existence of conic K"ahler-Einstein metrics for angles close to the critical limit

Identification of Tian-Yau metrics as limits of degenerating conic metrics

Analysis of metric behavior at various scales as the cone angle approaches the limit

Abstract

Let $X$ be a Fano manifold of dimension at least $2$ and $D$ be a smooth divisor in a multiple of the anticanonical class, $\frac1\alpha(-K_X)$ with $\alpha>1$. It is well-known that K\"ahler-Einstein metrics on $X$ with conic singularities along $D$ may exist only if the angle $2\pi\beta$ is bigger than some positive limit value $2\pi\beta_*$. Under the hypothesis that the automorphisms of $D$ are induced by the automorphisms of the pair $(X,D)$, we prove that for $\beta>\beta_*$ close enough to $\beta_*$, such K\"ahler-Einstein metrics do exist. We identify the limits at various scales when $\beta\rightarrow\beta_*$ and, in particular, we exhibit the appearance of the Tian-Yau metric of $X\setminus D$.