A remark on the Donaldson-Futaki invariant for sequences of test configurations

TL;DR

This paper derives an explicit formula for the Donaldson-Futaki invariant for sequences of test configurations compatible with a Kähler metric on a polarized algebraic manifold, aiding in stability analysis.

Contribution

It provides a new explicit formula for the Donaldson-Futaki invariant for sequences of test configurations, enhancing understanding of stability in Kähler geometry.

Findings

Explicit formula for the Donaldson-Futaki invariant $F_1$ for sequences of test configurations.

Improved tools for analyzing K-stability of polarized algebraic manifolds.

Facilitates future research in algebraic and differential geometry stability criteria.

Abstract

In this note, we consider a sequence of test configurations compatible with a Kaehler metric in $c_1(L)$ on a polarized algebraic manifold $(X,L)$. Then an explicit formula for the Donaldson-Futaki invariant $F_1$ for the sequence will be given.