A non-Archimedean approach to K-stability

TL;DR

This paper introduces a non-Archimedean geometric framework to analyze K-stability of smooth Fano varieties, providing new proofs and connections to valuative criteria without relying on the Minimal Model Program.

Contribution

It develops a non-Archimedean approach to K-stability, proving equivalence with Ding stability and recovering valuative criteria through psh metrics and the Legendre transform.

Findings

K-stability is equivalent to Ding stability via non-Archimedean methods.

The valuative criterion of K-stability is recovered using psh metrics.

Results apply to arbitrary smooth polarized varieties without the Minimal Model Program.

Abstract

We study K-stability properties of a smooth Fano variety X using non-Archimedean geometry, specifically the Berkovich analytification of X with respect to the trivial absolute value on the ground field. More precisely, we view K-semistability and uniform K-stability as conditions on the space of plurisubharmonic (psh) metrics on the anticanonical bundle of X. Using the non-Archimedean Calabi-Yau theorem and the Legendre transform, this allows us to give a new proof that K-stability is equivalent to Ding stability. By choosing suitable psh metrics, we also recover the valuative criterion of K-stability by Fujita and Li. Finally, we study the asymptotic Fubini-Study operator, which associates a psh metric to any graded filtration (or norm) on the anticanonical ring. Our results hold for arbitrary smooth polarized varieties, and suitable adjoint/twisted notions of K-stability and Ding stability. They do not rely on the Minimal Model Program.