  In this paper we compare different notions of transversality for possible
singular complex algebraic or analytic subsets of an ambient complex manifold
and prove a refined intersection formula for their Chern-Schwartz-MacPherson
classes. In case of a transversal intersection of complex Whitney stratified
sets, this result is well known. For splayed subsets it was conjectured (and
proven in some cases) by Aluffi and Faber. Both notions are stronger than a
micro-local "non-characteristic intersection" condition for the characteristic
cycles of (associated) constructible functions, which nevertheless is enough to
imply the asked refined intersection formula for the Chern-Schwartz-MacPherson
classes. The proof is based the multiplicativity of Chern-Schwartz-MacPherson
classes with respect to cross products, as well as a new Verdier-Riemann-Roch
theorem for "non-characteristic pullbacks".
