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

TL;DR

This paper proves that for asymptotically conical K"ahler-Ricci shrinkers with decaying Ricci curvature, the existence of such a metric implies K-polystability of the associated polarized Fano fibration, supporting a conjecture linking geometry and stability.

Contribution

It establishes one direction of the Sun-Zhang conjecture, showing that existence of a K"ahler-Ricci shrinker implies K-polystability under decay conditions.

Findings

Existence of a K"ahler-Ricci shrinker implies K-polystability.

Supports the conjecture linking geometric existence to algebraic stability.

Focuses on asymptotically conical shrinkers with decaying Ricci curvature.

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.