  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\'er and Rim\'anyi.
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.
