A valuative criterion of K-polystability

TL;DR

This paper introduces a new computable invariant for log Fano pairs with torus actions, providing a criterion for K-polystability and demonstrating applications to Fano varieties with g-solitons.

Contribution

It develops a novel invariant that characterizes K-polystability for log Fano pairs with torus actions, enabling explicit checks and applications.

Findings

Invariant is greater than one iff the pair is K-polystable

Examples of Fano varieties with g-solitons are constructed

Provides a practical criterion for stability in algebraic geometry

Abstract

For any log Fano pair with a torus action, we associate a computable invariant to it, such that the pair is (weighted) K-polystable if and only if this invariant is greater than one. As an application, we present examples of Fano varieties admitting $g$-solitons for any weight function $g$.