Localizing the Donaldson-Futaki invariant

TL;DR

This paper employs equivariant localization to relate the Donaldson-Futaki invariant of certain test configurations to the Futaki invariant of the central fiber, providing a localized approach to understanding stability in Kähler geometry.

Contribution

It establishes a localization formula for the Donaldson-Futaki invariant, connecting it to the Futaki invariant of the central fiber in smooth or orbifold cases.

Findings

Proves the equivalence of Donaldson-Futaki and Futaki invariants via localization.

Extends localization techniques to the deformation to the normal cone.

Provides tools for analyzing stability in Kähler geometry.

Abstract

We use the equivariant localization formula to prove that the Donaldson-Futaki invariant of a compact smooth (K{\"a}hler) test configuration coincides with the Futaki invariant of the induced action on the central fiber when this fiber is smooth or have orbifold singularities. We also localize the Donaldson-Futaki invariant of the deformation to the normal cone.