Poincar\'e-Hopf Theorem for Isolated Determinantal Singularities

TL;DR

This paper generalizes the Poincaré-Hopf theorem to projective varieties with isolated determinantal singularities using new index concepts.

Contribution

It introduces two generalized Poincaré-Hopf indices and proves a related theorem for varieties with determinantal singularities.

Findings

Established a Poincaré-Hopf type theorem for such varieties

Developed two new generalizations of the Poincaré-Hopf index

Applied the theory to projective varieties with isolated determinantal singularities

Abstract

Let $X \subset\mathbb{P}^r$ be a projective $d$-variety with isolated determinantal singularities and $\omega$ be a $1$-form on $X$ with a finite number of singularities (in the stratified sense). Under some technical conditions on $r$ we use two generalization of Poincar\'e-Hopf index with the goal of proving a Poincar\'e-Hopf Type Theorem for $X$.