Local Euler obstructions of reflexive projective varieties

Abstract

The local Euler obstruction and the polar multiplicities are key ingredients in the study of the local topology of stratified spaces. Despite their importance, in general it's very difficult to compute them. In this paper we introduce the concept of reflexive projective varieties. These are stratified projective varieties with certain dimension constraints on their dual varieties. We prove that for such varieties, their local Euler obstructions and polar multiplicities are completely determined by their Chern-Schwartz-MacPherson classes. Explicit formulas are presented, based on which we also propose an algorithm to compute such geometric invariants using the input of the characteristic classes. These characteristic classes are easier to compute in practice and may be carried out by computer algebra. In the case of reflexive group orbits, the formula and the algorithm are further refined.

Our method is purely algebraic and works for arbitrary algebraically closed field of characteristic 0. As examples we compute the local Euler obstructions of ordinary determinantal varieties to illustrate our method.