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We present an analytic proof of the relationship between the Calabi-Futakiinvariant for a K\"ahler manifold relative to a holomorphic vector field with anondegenerate 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 ahypothesis regarding the normal form of the vector field near its zero. Analgebro-geometric proof was given by Sz\'ekelyhidi by reducing the situation tothe case of projective manifolds for rational data and using Donaldson-Futakiinvariants. Our proof will be an application of degenerate localization.

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