On log K-stability for asymptotically log Fano varieties

TL;DR

This paper proves that certain asymptotically log Fano varieties cannot admit Kähler-Einstein edge metrics with small angles, confirming a conjecture and linking stability conditions to metric existence.

Contribution

It establishes a non-existence result for Kähler-Einstein edge metrics on a class of asymptotically log Fano varieties, confirming a conjecture by Cheltsov and Rubinstein.

Findings

Asymptotically log Fano varieties with irreducible divisor D and big -K_X-D do not admit KE edge metrics for small angles.

Confirms the conjecture of Cheltsov and Rubinstein regarding log K-stability and metric existence.

Provides conditions under which KE edge metrics cannot exist on asymptotically log Fano varieties.

Abstract

The notion of asymptotically log Fano varieties was given by Cheltsov and Rubinstein. We show that, if an asymptotically log Fano variety $(X, D)$ satisfies that $D$ is irreducible and $-K_X-D$ is big, then $X$ does not admit K\"ahler-Einstein edge metrics with angle $2\pi\beta$ along $D$ for any sufficiently small positive rational number $\beta$. This gives an affirmative answer to a conjecture of Cheltsov and Rubinstein.