A theorem of Poincar\'e-Hopf type

TL;DR

This paper extends the Poincaré-Hopf theorem to complex sheaves with constructible cohomology, providing a stratified formula that relates indices to the Euler characteristic.

Contribution

It introduces a stratified Poincaré-Hopf formula for complexes of sheaves, generalizing classical results to a broader algebraic setting.

Findings

Derived an algebraic formula for the Euler characteristic of sheaf complexes.

Established a stratified Poincaré-Hopf theorem based on smooth case and additivity.

Defined a suitable notion of index for sheaf complexes.

Abstract

We compute (algebraically) the Euler characteristic of a complex of sheaves with constructible cohomology. A stratified Poincar\'e-Hopf formula is then a consequence of the smooth Poincar\'e-Hopf theorem and of additivity of the Euler-Poincar\'e characteristic with compact supports, once we have a suitable definition of index.