Localizing the Donaldson-Futaki invariant

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.