Uniform K-stability and plt blowups of log Fano pairs

TL;DR

This paper explores the connection between uniform K-stability and plt blowups of log Fano pairs, providing criteria for stability, analyzing covers, and offering a new proof for the K-semistability of the projective plane.

Contribution

It establishes that evaluating specific volume-based invariants on plt blowups suffices to determine uniform K-stability of log Fano pairs, and discusses stability under crepant covers.

Findings

Uniform K-stability can be tested via invariants on plt blowups.

Uniform K-stability of certain pairs is preserved under crepant finite covers.

A new proof of K-semistability of the projective plane is provided.

Abstract

We show relationships between uniform K-stability and plt blowups of log Fano pairs. We see that it is enough to evaluate certain invariants defined by volume functions for all plt blowups in order to test uniform K-stability of log Fano pairs. We also discuss the uniform K-stability of two log Fano pairs under crepant finite covers. Moreover, we give another proof of K-semistability of the projective plane.