A decomposition formula for J-stability and its applications

TL;DR

This paper introduces a new decomposition formula for J-stability, linking non-archimedean energy to intersection numbers, and explores its implications for algebraic geometry and stability notions.

Contribution

It provides a novel decomposition formula for J-energy and establishes equivalences of stability notions for surfaces, along with applications to K-stability of minimal surfaces.

Findings

Decomposition formula expresses J-energy via intersection numbers.

Equivalence of slope J-stability and J-stability for surfaces.

Proof of uniform K-stability for minimal surfaces.

Abstract

For algebro-geometric study of J-stability, a variant of K-stability, we prove a decomposition formula of non-archimedean $\mathcal{J}$-energy of $n$-dimensional varieties into $n$-dimensional intersection numbers rather than $(n+1)$-dimensional ones, and show the equivalence of slope $\mathrm{J}^H$-(semi)stability and $\mathrm{J}^H$-(semi)stability for surfaces when $H$ is pseudoeffective. Among other applications, we also give a purely algebro-geometric proof of a uniform K-stability of minimal surfaces due to [23], and provides examples which are J-stable (resp., K-stable) but not uniformly J-stable (resp., uniformly K-stable).