Asymptotic slopes of the Aubin-Yau functional and calculation of the Donaldson-Futaki invariant

TL;DR

This paper provides an explicit formula for the asymptotic slope of the Aubin-Yau functional on complex surfaces, linking it to the Donaldson-Futaki invariant, thus aiding in K-stability checks.

Contribution

It extends previous work by deriving a formula for the Aubin-Yau slope on surfaces, connecting it explicitly to the Donaldson-Futaki invariant and K-stability analysis.

Findings

Explicit formula for Aubin-Yau slope on surfaces

Connection between slope and Donaldson-Futaki invariant

Method to check K-stability directly

Abstract

We derive an explicit formula for the asymptotic slope of the Aubin-Yau functional along a Bergman geodesic on a surface of complex dimension 2, extending the work of Phong-Sturm on Riemann surfaces. This is equivalent to an explicit calculation of the Donaldson-Futaki invariant of a test configuration. The slope is given as a rational linear combination of period integrals of rational functions that sum to a rational number. The result gives a way to check directly whether a two dimensional projective variety is K-stable.