Local Euler Obstructions of Reflective Projective Varieties

TL;DR

This paper introduces reflective projective varieties, showing their Chern-Schwartz-MacPherson classes determine local Euler obstructions and polar degrees, with an algorithm for computation and examples including quadratic hypersurfaces.

Contribution

It defines reflective projective varieties and establishes a method to compute local Euler obstructions from their characteristic classes, with practical algorithms and examples.

Findings

Chern-Schwartz-MacPherson classes determine local Euler obstructions for reflective varieties.

An algorithm is provided for computing local Euler obstructions of group orbit varieties.

Explicit computations for quadratic hypersurfaces and determinantal varieties are presented.

Abstract

In this note we introduce the concept of reflective projective varieties. These are stratified projective varieties with certain dimension constraints on their dual varieties. We prove that for such varieties, the Chern-Schwartz-MacPherson classes of the strata completely determine the local Euler obstructions and the polar degrees. We also propose an algorithm to compute the local Euler obstructions when such varieties are formed by group orbits. As examples we compute the local Euler obstructions of quadratic hypersurfaces and ordinary determinantal varieties to illustrate our method.