The Futaki Invariant of K\"ahler Blowups with Isolated Zeros via Localization

TL;DR

This paper provides an analytic proof relating the Futaki invariant of a K"ahler manifold to that of its blowup at an isolated zero, extending previous results and clarifying assumptions using localization techniques.

Contribution

It introduces an analytic localization-based proof of the Futaki invariant relationship for blowups with isolated zeros, extending prior work on K"ahler surfaces.

Findings

Established the relationship between Futaki invariants of a manifold and its blowup.

Extended results to cases with isolated zeros on the exceptional divisor.

Clarified the normal form hypothesis near zeros.

Abstract

We present an analytic proof of the relationship between the Calabi-Futaki invariant for a K\"ahler manifold relative to a holomorphic vector field with a nondegenerate zero and the corresponding invariant of its blowup at that zero, restricting to the case that zeros on the exceptional divisor are isolated. This extends the results of Li and Shi for K\"ahler surfaces. We also clarify a hypothesis regarding the normal form of the vector field near its zero. An algebro-geometric proof was given by Sz\'ekelyhidi by reducing the situation to the case of projective manifolds for rational data and using Donaldson-Futaki invariants. Our proof will be an application of degenerate localization.