On indices of 1-forms on determinantal singularities

TL;DR

This paper extends the concept of indices for 1-forms to essentially isolated determinantal singularities, exploring their properties and relations to classical indices like the Poincaré-Hopf and radial indices.

Contribution

It introduces analogues of the Poincaré-Hopf index for 1-forms on determinantal singularities and examines their properties and interrelations.

Findings

Defined indices for 1-forms on determinantal singularities

Established relations between these indices and the radial index

Analyzed properties of the homological index for isolated cases

Abstract

We consider 1-forms on, so called, essentially isolated determinantal singularities (a natural generalization of isolated ones), show how to define analogues of the Poincar\'e--Hopf index for them, and describe relations between these indices and the radial index. For isolated determinantal singularities, we discuss properties of the homological index of a holomorphic 1-form and its relation with the Poincar\'e--Hopf index.