Chern classes and characteristic cycles of determinantal varieties

Xiping Zhang

Let K be an algebraically closed field of characteristic 0. For m
\ge n, we define tau_{m,n,k} to be the set of mxn matrices over K
with kernel dimension \ge k. This is a projective subvariety of
P^{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=C, 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}-tau_{m,n,k+1}.

The irreducibility of
the characteristic cycle of the intersection cohomology sheaf
follows from the Kashiwara-Subson'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}.