Positivity of Segre-MacPherson classes

Paolo Aluffi, Leonardo C. Mihalcea, Jörg Schürmann, Changjian Su

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 Feh{\'e}r and Rim{\'a}nyi. 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.