A Poincar\'e-Hopf Type Formula for A Pair of Vector Fields

TL;DR

This paper extends a Poincaré-Hopf type formula to handle pairs of vector fields with non-isolated singularities on smooth manifolds, broadening the applicability of topological invariants in complex geometry.

Contribution

It generalizes existing formulas to include non-isolated singularities using a special connection and Chern character, advancing the understanding of vector field zeros.

Findings

Established a Poincaré-Hopf type formula for non-isolated singularities.

Constructed a special connection to analyze Chern character differences.

Extended previous results to broader classes of vector fields and singularities.

Abstract

For two complex vector bundles admitting a homomorphism between them, a Poincar\'e-Hopf formula for the difference of the Chern character numbers of these two vector bundles with isolated singularities is established by Huitao Feng, Weiping Li and Weiping Zhang. This article extend their reslut about Poincar\'e-Hopf type formula for the difference of the Chern character numbers to the non-isolated singularities. A special connection $\widetilde{\nabla}_{1}^{E}$ be constructed, the Chern character ${\rm ch}(E,\widetilde{\nabla}_{1}^{E})$ is the key role for the non-isolated singularities. As a consequence, a Poincar\'e-Hopf type formula for a pair of vector field with the function $h^{T_{\mathbb{C}}M}(\cdot,\cdot)$ has non-isolated zero points over a closed, oriented smooth manifold of dimension $2n$ is established.