Local Euler obstructions for determinantal varieties

TL;DR

This paper derives formulas for local Euler obstructions of determinantal varieties using invariant de Rham complexes and character formulas for equivariant D-modules, connecting algebraic and geometric methods.

Contribution

It provides a unified derivation of Euler obstruction formulas for various determinantal varieties, extending and clarifying previous implicit results.

Findings

Formulas for local Euler obstructions of determinantal varieties derived explicitly.

Connections established between invariant D-modules and characteristic cycles.

Results align with and extend previous work by Boe, Fu, Gaffney, Grulha, Ruas, Promtapan, Rimányi, and Zhang.

Abstract

The goal of this note is to explain a derivation of the formulas for the local Euler obstructions of determinantal varieties of general, symmetric and skew-symmetric matrices, by studying the invariant de Rham complex and using character formulas for simple equivariant $D$-modules. These calculations are then combined with standard arguments involving Kashiwara's local index formula and the description of characteristic cycles of simple equivariant $D$-modules. The formulas are implicit in the work of Boe and Fu, and in the case of general matrices they have also been obtained recently by Gaffney--Grulha--Ruas, for skew-symmetric matrices by Promtapan and Rim\'anyi, and for all cases by Zhang.