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 extends and unifies several previous results in the literature, and yields several new results. For example, we prove that Behrend's Donaldson-Thomas invariant for a closed subvariety of an abelian variety is effective; that the intersection homology Chern class of the theta divisor for a non-hyperellptic curve is signed-effective; and we prove more 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. Finally, 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.