Chern classes and Characteristic Cycles of Determinantal Varieties

TL;DR

This paper computes explicit formulas for characteristic classes and cycles of determinantal varieties, revealing their geometric and topological properties, and explores their implications in algebraic geometry.

Contribution

It provides the first explicit formulas for Chern-Mather, Chern-Schwartz-MacPherson classes, and characteristic cycles of determinantal varieties, including their conormal cycles and Euclidean Distance degrees.

Findings

Formulas for Chern-Mather and Chern-Schwartz-MacPherson classes.

Explicit computation of characteristic cycles and conormal cycles.

Confirmation of irreducibility of characteristic cycles for intersection cohomology sheaves.

Abstract

Let $K$ be an algebraically closed field of characteristic $0$. For $m\geq n$, we 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 $\bbP^{mn-1}$, and is called the (generic) determinantal variety. In most cases $\tau_{m,n,k}$ is singular with singular locus $\tau_{m,n,k+1}$. In this paper we give explicit formulas computing the Chern-Mather class ($c_M$) and the Chern-Schwartz-MacPherson class ($c_{SM}$) of $\tau_{m,n,k}$, as classes in the projective space. We also obtain formulas for the conormal cycles and the characteristic cycles of these varieties, and for their generic Euclidean Distance degree. Further, when $K=\bbC$, we prove that the characteristic cycle of the intersection cohomology sheaf of a determinantal variety agrees with its conormal cycle (and hence is irreducible). Our formulas are based on calculations of degrees of certain Chern classes of the universal bundles over the Grassmannian. For some small values of $m,n,k$, we use Macaulay2 to exhibit examples of the Chern-Mather classes, the Chern-Schwartz-MacPherson classes and the classes of characteristic cycles of $\tau_{m,n,k}$. On the basis of explicit computations in low dimensions, we formulate conjectures concerning the effectivity of the classes and the vanishing of specific terms in the Chern-Schwartz-MacPherson classes of the largest strata $\tau_{m,n,k}\smallsetminus \tau_{m,n,k+1}$. The irreducibility of the characteristic cycle of the intersection cohomology sheaf follows from the Kashiwara-Dubson's microlocal index theorem, a study of the `Tjurina transform' of $\tau_{m,n,k}$, and the recent computation of the local Euler obstruction of $\tau_{m,n,k}$ .