[Paper Review] Remarks on Localizing Futaki-Morita Integrals At Isolated Degenerate Zeros

Luke Cherveny|arXiv (Cornell University)|Nov 15, 2016

Geometry and complex manifolds6 references3 citations

TL;DR

This paper extends Bott's localization theorem to holomorphic vector fields with isolated degenerate zeros by deriving a closed-form formula for Futaki-Morita integrals using Grothendieck residues and power series expansions. The key contribution is a single, explicit formula for the invariant at a maximally degenerate zero on $\mathbb{C}P^n$, eliminating summation over fixed points and recovering known results in special cases.

ABSTRACT

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 Kähler blowup.

Motivation & Objective

• To generalize Bott’s localization theorem to holomorphic vector fields with isolated degenerate zeros, where the standard nondegenerate assumption fails.
• To eliminate the need for summation over multiple fixed points in Futaki-Morita integral computations by deriving a single, global formula.
• To provide a streamlined, self-contained exposition of localization techniques in complex geometry, particularly for degenerate zero cases.
• To lay the foundation for computing Futaki invariants on blowups of Kähler manifolds, where degenerate zeros naturally arise.
• To demonstrate the formula’s effectiveness via explicit computation on $\mathbb{C}P^n$ with a maximally degenerate vector field.

Proposed method

• Apply Bott’s transgression argument to the Futaki-Morita integral using a power series expansion of the curvature and endomorphism terms.
• Use the Grothendieck residue representation via the Bochner-Martinelli kernel to express the integral as a residue at the isolated degenerate zero.
• Derive a general formula (Theorem 1.3) by computing higher-order partial derivatives of $\phi(DX) \det B$ at the origin, where $B$ encodes the local zero structure.
• Construct the matrix $B$ explicitly from the vector field’s local equations $z_i^{\alpha_i+1} = \sum B_{ij} X_j$, ensuring the existence via the strong Hilbert Nullstellensatz.
• Verify the formula by applying it to a maximally degenerate holomorphic vector field on $\mathbb{C}P^n$, using recursive factoring of monomials.
• Utilize determinant identities and derivative evaluations to simplify the final expression, isolating non-vanishing terms at $z=0$.

Experimental results

Research questions

• RQ1 Can the Futaki-Morita integral be localized at isolated degenerate zeros without summing over multiple fixed points?
• RQ2 What is the precise form of the localization formula when the zero is maximally degenerate, such as on $\mathbb{C}P^n$?
• RQ3 How does the Grothendieck residue formalism extend to degenerate zero loci in the context of holomorphic vector fields?
• RQ4 Can the formula recover known invariants like the Euler characteristic and first Chern class integral on $\mathbb{C}P^n$?
• RQ5 What is the behavior of the Futaki invariant under blowups when the lifted vector field has degenerate zeros in the exceptional divisor?

Key findings

• A new localization formula (Theorem 1.3) expresses the Futaki-Morita integral as a single, closed-form derivative expression at a degenerate zero, avoiding summation over fixed points.
• For the maximally degenerate vector field on $\mathbb{C}P^n$, the formula yields ${n+k \choose n} f_\phi(X) = \frac{(-1)^{n+k}}{n!} \left( \frac{\partial^n \phi(DX)}{\partial z_1^n} + \sum_{j=2}^n \frac{\partial}{\partial z_1^n \partial z_j} (\phi(DX) \cdot z_1^j) \right) \big|_{z=0}$, as shown in Proposition 4.1.
• The Euler characteristic of $\mathbb{C}P^n$ is correctly computed as $n+1$, confirming the formula’s consistency with known topology.
• The first Chern class integral $\int_{\mathbb{C}P^n} c_1^n$ evaluates to $(n+1)^n$, matching the expected value from Chern-Weil theory.
• The Futaki invariant vanishes for $\phi(A) = [\operatorname{Tr}(A)]^{n+1}$, consistent with the Kähler-Einstein property of $\mathbb{C}P^n$ under the Fubini-Study metric.
• The formula also correctly gives zero for $\phi(A) = \operatorname{Tr}(A) \det A$, confirming a known vanishing result of Futaki for such invariants.