The Donaldson-Futaki invariant for sequences of test configurations

TL;DR

This paper introduces a new way to measure stability of polarized algebraic manifolds using sequences of test configurations, linking it to the existence of constant scalar curvature Kähler metrics.

Contribution

It defines the Donaldson-Futaki invariant for sequences of test configurations and establishes a stronger notion of K-stability.

Findings

Strong K-semistability implies existence of constant scalar curvature Kähler metrics

New invariant extends classical Donaldson-Futaki invariant to sequences

Provides a criterion for K-stability based on sequences of test configurations

Abstract

In this note, given a polarized algebraic manifold $(X,L)$, we define the Donaldson-Futaki invariant for a sequence of test configurations for $(X,L)$ with exponents tending to infinity. This then allows us to define a strong version of K-stability or K-semistability for $(X,L)$. In particular, $(X,L)$ will be shown to be K-semistable in this strong sense if the polarization class $c_1(L)$ admits a constant scalar curvature Kaehler metric.