K-polystability of Asymptotically Conical Kähler-Ricci Shrinkers

Abstract

Recently, Sun-Zhang have developed an algebraic theory for Kähler-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 Kähler-Ricci shrinker metric is equivalent to a notion of K-stability. We prove one direction of this conjecture, namely that existence of a Kähler-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. As an application, we give a non-existence result: if $M$ is the blowup of a six-dimensional quadric along a two-dimensional subquadric, then the total space $X$ of the cube root of $K_M$ is a polarized Fano fibration not admitting a Kähler-Ricci shrinker.

Added section 7. 45 pages