Poincar\'e-Hopf Theorem for Isolated Determinantal Singularities

TL;DR

This paper extends the Poincaré-Hopf theorem to projective algebraic varieties with isolated determinantal singularities using generalized indices of 1-forms.

Contribution

It introduces a Poincaré-Hopf type theorem for varieties with determinantal singularities employing new index generalizations.

Findings

Proves a Poincaré-Hopf type theorem for such varieties.

Utilizes two generalizations of the Poincaré-Hopf index.

Establishes a relation between indices and singularities.

Abstract

Let $X$ be a projective algebraic $d$-variety endowed with isolated determinantal singularities, and let $\omega$ be a $1$-form on $X$ exhibiting a finite number of singularities (in the stratified sense). Under some technical conditions, we use two generalizations of Poincar\'e-Hopf index with the goal of proving a Poincar\'e-Hopf Type Theorem for $X$.