2017/06/07 by Xiping Zhang, Zhang, Xiping · 1 citation
Mathematics · #14C17 #Advanced Algebra and Geometry #Advanced Combinatorial Mathematics #Algebraic Geometry (math.AG) #Algebraic Geometry and Number Theory #FOS: Mathematics
paper · pdf · doi:10.48550/arxiv.1706.02032
openalex publication_date 2017/06/07 · openalex created_date 2017/06/15 · openalex updated_date 2026/07/28
For m≥ n, Let K be an algebraic closed base field, and define τm,n,k to be the set of m× n matrices over K with kernel dimension ≥ k. This is a projective subvariety of ℙmn-1, and is usually called determinantal variety. In most cases τm,n,k is singular with singular locus τm,n,k+1. In this paper we compute the local Euler obstruction of τm,n,k, and we prove that the characteristic cycle of the intersection cohomology complex of τm,n,k is irreducible. We also give an explicit formula for the Chern-Mather class of τm,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 τm,n,k for some small values of m,n,k. Over the complex numbers, the local Euler obstruction of τm,n,k was recently computed by N.~Grulha, T.~Gaffney and M.~Ruas by methods in complex geometry.