It is known that cup product is Poincaré dual to the intersection. I'm referring to the following fact: if $X$ is a closed, oriented smooth manifold and $A, B$ are transverse-intersecting oriented submanifolds of codimension $i, j$ respectively, then
$$[A \cap B]^* = [A]^* \smile [B]^* \in H^{i+j}(X)\space ,$$
where the asterisk denotes Poincaré dual.
My question: is the same true if we take $A, B$ to be transverse-intersecting algebraic varieties? (And does that even make sense? I think that an algebraic subvariety defines an homology class given by the pushforward of the inclusion of the top class, and therefore it makes sense; but correct me if I'm wrong).
For context: I'm studying Schubert calculus, and I want to use this fact when $A, B$ are Schubert varieties, but I think Schubert varieties aren't smooth manifolds in general, since they contain singular points.
This is basically the approach via stratifolds developed by Kreck; see
and