A Poincar\'e-Hopf type formula for Chern character numbers

TL;DR

This paper extends the Poincaré-Hopf index formula to complex vector bundles with isolated singularities, providing a new formula for the difference in their Chern character numbers and applying it to complex vector fields.

Contribution

It introduces a Poincaré-Hopf type formula for Chern character differences in complex vector bundles with isolated singularities, expanding classical index theory.

Findings

Derived a new formula relating Chern character numbers and singularities.

Extended Poincaré-Hopf index formula to complex vector fields.

Applicable to complex vector bundles with isolated singularities.

Abstract

For two complex vector bundles admitting a homomorphism with isolated singularities between them, we establish a Poincar\'e-Hopf type formula for the difference of the Chern character numbers of these two vector bundles. As a consequence, we extend the original Poincar\'e-Hopf index formula to the case of complex vector fields (to appear in Mathematische Zeitschrift)