K-polystability of Asymptotically Conical K\"ahler-Ricci Shrinkers

Abstract

Recently, Sun-Zhang have developed an algebraic theory for K\"ahler-Ricci shrinkers showing that they admit the structure of a polarized Fano fibration $(\pi: X \to Y, \xi)$. In particular, they conjecture that existence of a K\"ahler-Ricci shrinker metric is equivalent to a notion of K-stability. We prove one direction of this conjecture, namely that existence of a K\"ahler-Ricci shrinker metric $g$ implies K-polystability of $(\pi: X \to Y, \xi)$, in the case that the Ricci curvature of $g$ decays at infinity.