[Paper Review] The Woods Hole trace formula and indices for vector fields and foliations on $\mathbb{C}^2$

Valente Ramírez|arXiv (Cornell University)|Aug 18, 2016

Advanced Differential Equations and Dynamical Systems2 references3 citations

TL;DR

This paper demonstrates that the Woods Hole trace formula unifies and generalizes key index theorems for vector fields and foliations on ℂ², including the Euler-Jacobi, Baum-Bott, and Camacho-Sad formulas. By constructing a natural endomorphism $f_v$ on $\mathbb{P}^2$ associated to a polynomial vector field $v$, the authors derive these classical index theorems as particular cases of the Woods Hole formula, providing a systematic framework to discover new index relations for singular foliations and vector fields.

ABSTRACT

The purpose of this work is to introduce the reader to the Woods Hole trace formula and to show how several well-known index theorems for foliations, vector fields and endomorphisms can be rephrased as particular cases of such formula. The main objective in doing this is to use the Woods Hole formula in the future to produce new index theorems.

Motivation & Objective

• To show that the Woods Hole trace formula subsumes and generalizes known index theorems for vector fields and foliations on ℂ².
• To establish a geometric construction of a holomorphic endomorphism $f_v$ on $\mathbb{P}^2$ from a polynomial vector field $v$ on $\mathbb{C}^2$ with non-degenerate singularities.
• To use the Woods Hole formula applied to specific sheaves and lifts to recover classical index formulas such as the Euler-Jacobi, Baum-Bott, and Camacho-Sad theorems.
• To lay a foundation for discovering new index relations by leveraging the Woods Hole formula as a unifying framework.

Proposed method

• The Woods Hole trace formula is applied to coherent analytic sheaves on $\mathbb{P}^2$, using a holomorphic endomorphism $f_v$ naturally associated to a polynomial vector field $v$ of degree $d$.
• For the first Euler-Jacobi relation, the formula is applied to the ideal sheaf $\mathcal{I}_L$ of the line at infinity $L$, with a natural inclusion $\varphi = \iota: f^*\mathcal{I}_L \to \mathcal{I}_L$.
• For the second Euler-Jacobi relation, the formula is applied to $\mathscr{F} = \Omega^1_{\mathbb{P}^2} \otimes \mathcal{I}_L$, with the lift $\varphi = df \otimes \iota$.
• The Camacho-Sad formula is recovered by restricting the formula to the line $L$, using the conormal sheaf $\mathcal{N}^*_L$ and the differential $df_\xi$ as the lift.
• The endomorphism $f_v$ is explicitly constructed via homogenization of the vector field components, ensuring $f_v$ preserves $L$ and has isolated fixed points when $v$ is non-dicritical and non-degenerate.
• The key insight is that the linearization $Dv(p)$ of $v$ at a finite singularity satisfies $Dv(p) = Df_v(p) - I$, linking the vector field dynamics to the endomorphism's Jacobian.

Experimental results

Research questions

• RQ1 Can the Woods Hole trace formula be used to systematically derive and unify known index theorems for vector fields and foliations on ℂ²?
• RQ2 What is the precise geometric construction of a holomorphic endomorphism $f_v$ on $\mathbb{P}^2$ associated to a polynomial vector field $v$ on $\mathbb{C}^2$?
• RQ3 How do the classical index theorems—Euler-Jacobi, Baum-Bott, and Camacho-Sad—emerge as special cases of the Woods Hole formula?
• RQ4 What new index relations can be discovered by extending this framework beyond the known theorems?

Key findings

• The first Euler-Jacobi relation $\sum_{v(p)=0} \frac{1}{\det(Dv(p))} = 0$ is derived by applying the Woods Hole formula to the ideal sheaf $\mathcal{I}_L$ with the inclusion $\varphi = \iota$.
• The second Euler-Jacobi relation $\sum_{v(p)=0} \frac{\operatorname{tr}(Dv(p))}{\det(Dv(p))} = 0$ follows from the Woods Hole formula applied to $\mathscr{F} = \Omega^1_{\mathbb{P}^2} \otimes \mathcal{I}_L$ with the lift $\varphi = df \otimes \iota$.
• The Camacho-Sad formula $\sum_{L \cap \operatorname{Sing}\mathcal{F}_v} \operatorname{CS}(\mathcal{F}_v, L, p) = 1$ is recovered by restricting the Woods Hole formula to the line $L$, using the conormal sheaf $\mathcal{N}^*_L$ and the differential $df_\xi$ as the lift.
• The Baum-Bott formula $\sum_{\operatorname{Sing}\mathcal{F}_v} \operatorname{BB}(\mathcal{F}_v, p) = (d+2)^2$ is shown to be a particular case of the Woods Hole formula via the sheaf $\Omega^1_{\mathbb{P}^2} \otimes \Omega^1_{\mathbb{P}^2}$ and the lift $df_\xi \otimes df_\xi$.
• The construction of $f_v$ via homogenization ensures that $f_v$ preserves the line at infinity $L$ and has isolated fixed points when $v$ is non-dicritical and has non-degenerate singularities.
• The Woods Hole formula provides a unifying framework: all known index theorems for polynomial vector fields on $\mathbb{C}^2$ are shown to be special cases of this single formula, suggesting a path to discovering new index relations.