Filtrations and test-configurations

TL;DR

This paper introduces a strengthened form of K-stability using filtrations, enabling the analysis of limits of test-configurations and linking stability to the existence of cscK metrics.

Contribution

It proposes a new stability notion based on filtrations, extending the framework of K-stability and connecting it to birational transformations and b-stability.

Findings

Manifolds with no automorphisms and cscK metrics satisfy the new stability.

The new stability notion considers limits of test-configurations.

Discussion of relations between this stability and b-stability.

Abstract

We introduce a strengthening of K-stability, based on filtrations of the homogeneous coordinate ring. This allows for considering certain limits of families of test-configurations, which arise naturally in several settings. We prove that if a manifold with no automorphisms admits a cscK metric, then it satisfies this stronger stability notion. We also discuss the relation with the birational transformations in the definition of b-stability.