Eigenvalues of vector fields, Bott's residue formula and integral invariants

TL;DR

This paper explores the eigenvalues of vector fields on almost-complex manifolds, connecting them with localization formulas and curvature integrals to identify obstructions to certain symmetries.

Contribution

It extends Bott's residue formula and Atiyah-Bott-Singer localization to relate eigenvalue multiplicities with geometric invariants and obstructions.

Findings

Eigenvalue multiplicities relate to zero point set structure.

A special case yields a vanishing-type result.

Obstructions to Killing vector fields are expressed via curvature integrals.

Abstract

Given a compatible vector field on a compact connected almost-complex manifold, we show in this article that the multiplicities of eigenvalues among the zero point set of this vector field have intimate relations. We highlight a special case of our result and reinterpret it as a vanishing-type result in the framework of the celebrated Atiyah-Bott-Singer localization formula. This new point of view, via the Chern-Weil theory and a strengthened version of Bott's residue formula observed by Futaki and Morita, can lead to an obstruction to Killing real holomorphic vector fields on compact Hermitian manifolds in terms of a curvature integral.