Positivity of Segre-MacPherson classes

Abstract

Let X be a complex nonsingular variety with globally generated tangent bundle. We prove that the signed Segre-MacPherson (SM) class of a constructible function on X with effective characteristic cycle is effective. This observation has a surprising number of applications to positivity questions in classical situations, unifying previous results in the literature and yielding several new results. We survey a selection of such results in this paper. For example, we prove general effectivity results for SM classes of subvarieties which admit proper (semi-)small resolutions and for regular or affine embeddings. Among these, we mention the effectivity of (signed) Segre-Milnor classes of complete intersections if X is projective and an alternation property for SM classes of Schubert cells in flag manifolds; the latter result proves and generalizes a variant of a conjecture of Fehér and Rimányi. Among other applications we prove the positivity of Behrend's Donaldson-Thomas invariant for a closed subvariety of an abelian variety and the signed-effectivity of the intersection homology Chern class of the theta divisor of a non-hyperelliptic curve; and we extend the (known) non-negativity of the Euler characteristic of perverse sheaves on a semi-abelian variety to more general varieties dominating an abelian variety.

Characteristic classes of symmetric and skew-symmetric degeneracy loci

“…The sign behavior of SSM classes is determined under very general circumstances in [AMSS2]. It would be interesting to check whether results in that paper imply our conjecture.…”

Mather classes of Schubert varieties via small resolutions

“…Sankaran and Vanchinathan [35] constructed IH-small resolutions of Bott-Samelson type for Grassmannian Schubert varieties in types D and C. Our goal of this paper is to express the coefficients of the Schubert expansion for the Chern-Mather class of Schubert varieties in even orthogonal Grassmannians OG(n, C 2n ) of Lie type D, as in the category of simply laced types, in terms of an integral involving Pfaffians and Sankaran and Vanchinathan's IH-small resolution. When it comes to types B and C, the expression we found from IH-small resolutions for Schubert varieties is for the Kazhdan-Lusztig class investigated by Aluffi, Mihalcea, Schuermann and Su [1,2] as well as Mihalcea and Singh [27]. Essentially, it turns out that Jones' outcomes for the Chern-Mather class coincides with the Kazhdan-Lusztig class [27, Page 15].…”

“…The Kazhdan-Lusztig class of Schubert varieties in isotropic or orthogonal Grassmannians can be signified as the pushforward of the total Chern class of tangent bundle c(T Z α ) of any IH-small resolution of singularity Z α , parallel to the Chern-Mather class as the pushforward of c(T Z α ) [2,23]. Since there is no explicit computation for the pushforward of the Chern class of the tangent bundle of the IH-small resolution of singularity except type A, we offer how to calculate it for the other classical types.…”

Structure constants for Chern classes of Schubert cells

“…cit. can be identified with the quotient of the CSM classes of the Schubert cells by the total Chern class of the flag variety, which are called the Segre-Schwartz-MacPherson (SSM) classes of the Schubert cells, see [4]. Under the non-degenerate Poincaré pairing on the equivariant cohomology of the flag variety, the CSM classes and SSM classes are dual to each other, just as the usual Schubert classes and the opposite ones.…”

“…If X is not smooth, we can embed X into a smooth ambient space, and use the total Chern class of the ambient space to define the SSM classes, see[4].…”