A note on the definition of K-stability

TL;DR

This paper discusses necessary modifications to the definition of K-stability, emphasizing the positivity of the Donaldson-Futaki invariant for certain test configurations, and clarifies the conditions for cscK manifolds.

Contribution

It refines the definition of K-stability by specifying the positivity condition of the Donaldson-Futaki invariant for non-trivial test configurations in codimension 2.

Findings

Revised the definition of K-stability.

Proved K-stability for cscK manifolds without holomorphic vector fields.

Clarified the role of test configurations in K-stability.

Abstract

As recently pointed out by Li and Xu, the definition of K-stability, and the author's proof of K-stability for cscK manifolds without holomorphic vector fields, need to be altered slightly: the Donaldson-Futaki invariant is positive for all test configurations which are not trivial in codimension 2.