Chern classes and characteristic cycles of determinantal varieties

Abstract

Abstract. Let K be an algebraically closed field of characteristic 0. For m ≥ n, we define τ m,n,k to be the set of m×n matrices over K with kernel dimension ≥ k. This is a projective subvariety of P mn−1 , and is called the (generic) determinantal variety. In most cases τ m,n,k is singular with singular locus τ 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 τ 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 τ 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 τ m,n,k τ 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 τ m,n,k , and the recent computation of the local Euler obstruction of τ m,n,k .

Motivic Chern classes and K‐theoretic stable envelopes

“…We prove formulas for the motivic Chern classes of the orbits, both in the traditional ‘localization’ form, and also expanded in the building blocks invented in Section 7. These results can be viewed as the K‐theory generalizations of [28, 47, 71].…”

Positivity of Segre-MacPherson classes

“…There are situations where the characteristic cycle associated to a constructible sheaf is known to be irreducible: examples include characteristic cycles of the intersection cohomology sheaves of Schubert varieties in the Grassmannian [BFL90], in more general minuscule spaces [BF97], of certain determinantal varieties [Zha18], and of the theta divisors in the Jacobian of a non-hyperelliptic curve [BB98]. In all such cases, š * (ϕ, X) is effective provided that T X is globally generated, by Theorem 2.2.…”

Mather classes of Schubert varieties via small resolutions

“…Beside our approach by IH-small resolutions and the advent of Pfaffians for types D, B and C, Mihalcea and Singh studied Mather classes from a resolution for the conormal spaces of cominuscule Schubert varieties in the equivariant setting [27]. We also refer to [32,38] for degeneracy loci of several types.…”