A remark on Li-Xu's pathology

TL;DR

This paper refines the Donaldson-Futaki invariant for test configurations, demonstrating that Li-Xu's pathology does not occur when considering the refined invariant, even for non-normal configurations with trivial normalization.

Contribution

It introduces a refined version of the Donaldson-Futaki invariant and shows it resolves the Li-Xu pathology in non-normal test configurations.

Findings

Refined invariant does not vanish for non-normal configurations.

Li-Xu's pathology is resolved with the refined invariant.

Non-normal test configurations with trivial normalization have non-zero refined invariant.

Abstract

For test configurations, the Donaldson-Futaki invariant F_1 is well-known. In this note, its refinement will be discussed. Then we see that Li-Xu's pathology doesn't occur, since their example of a non-normal test configuration, with trivial normalization, actually has non-vanishing F_1 in this refined sense.