Characteristic Classes of Homogeneous Essential Isolated Determinantal Varieties

Abstract

The (homogeneous) Essentially Isolated Determinantal Variety is the natural generalization of generic determinantal variety, and is fundamental example to study non-isolated singularities. In this paper we study the characteristic classes on these varieties. We give explicit formulas of their Chern-Schwartz-MacPherson classes via standard Schubert calculus. As corollaries we obtain formulas for their (generic) sectional Euler characteristics, characteristic cycles and polar classes. In particular, when such variety is a hypersurfaces we compute its Milnor class and the Euler characteristics of the local Milnor fibers. We prove that for such recursive group orbit hypersurfaces the local Euler obstructions completely determine the Milnor classes. In general for reflective group orbits, on the other hand we propose an algorithm to compute their local Euler obstructions via the Chern-Schwartz-MacPherson classes of the orbits, which can be obtained directly from representation theory. This builds a bridge from representation theory of the group action to the singularity theory of the induced orbits.