On log K-stability for 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.