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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.

We present an analytic proof of the relationship between the Calabi-Futaki invariant for a K��hler 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��hler 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��kelyhidi 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.

In this note we study the localization of Futaki-Morita integrals at isolated degenerate zeros by giving a streamlined exposition in the spirit of Bott and implement the localization procedure for a holomorphic vector field on $CP^n$ with a maximally degenerate zero, giving an essentially unique formula for the Futaki-Morita integral invariants without using a summation over multiple points. In a coming paper we will apply similar calculations to the Calabi-Futaki invariant of a K\"ahler blowup.

We study a generalization of Lian-Liu-Yau's notion of Euler data in genus zero and show that certain sequences of multiplicative equivariant characteristic classes on Kontsevich's stable map moduli with markings induce data satisfying the generalization. In the case of one or two markings, this data is explicitly identified in terms of hypergeometric type classes, constituting a complete extension of Lian-Liu-Yau's mirror principle in genus zero to the case of two marked points and establishing a program for the general case. We give several applications involving the Euler class of obstruction bundles induced by a concavex bundle on $P^n$., Comment: 35 pages