Local Euler Obstruction and Chern-Mather classes of Determinantal Varieties

Abstract

For $m\geq n$, Let $K$ be an algebraic closed base field, and define $\tau_{m,n,k}$ to be the set of $m\times n$ matrices over $K$ with kernel dimension $\geq k$. This is a projective subvariety of $\mathbb{P}^{mn-1}$, and is usually called determinantal variety. In most cases $\tau_{m,n,k}$ is singular with singular locus $\tau_{m,n,k+1}$. In this paper we compute the local Euler obstruction of $\tau_{m,n,k}$, and we prove that the characteristic cycle of the intersection cohomology complex of $\tau_{m,n,k}$ is irreducible. We also give an explicit formula for the Chern-Mather class of $\tau_{m,n,k}$ as a class in projective space. The irreducibility of the intersection cohomology characteristic cycle follows from the explicit computation of the local Euler obstruction, a study of the `Tjurina transforms' of determinantal varieties, and the Kashiwara-Dubson's microlocal index theorem. Our explicit formulas are based on calculations of degrees of certain Chern classes of the universal bundles over the Grassmannian. We use The Schubert 2 package in Macaulay2 to exhibit examples of the Chern-Mather class and the class of the characteristic cycle of $\tau_{m,n,k}$ for some small values of $m,n,k$. Over the complex numbers, the local Euler obstruction of $\tau_{m,n,k}$ was recently computed by N.~Grulha, T.~Gaffney and M.~Ruas by methods in complex geometry.