Intersection theory in analytic geometry

Question

This might be a weird/stupid question, but it came to me a couple of times, and I would like to get an answer for that.

In some papers I read, constantly the authors define some analytic subspaces, say $X$ and $Y$, and then the authors take the intersection product of their cycles $[X]\cdot [Y]$ in the homology group, without requiring $x$ intersect $Y$ transversely.

My question is, it seems that there is an intersection theory applied here, but I don't know how it works. Does Fulton-MacPherson's intersection theory directly apply to analytic setting ?

Answer 1

You don't say what kind of space $X$ and $Y$ are subspaces of. But if they sit in an oriented manifold there's an easy way to define an intersection product in homology. Namely if $M$ is an oriented $d$-manifold then there is a Poincaré duality isomorphism $$ H_i(M,\mathbf Z) \cong H^{d-i}_c(M,\mathbf Z)$$ between homology and compactly supported cohomology. The cohomology with compact support is a ring (though typically a non-unital ring) for the same reason that the usual cohomology is a ring: use contravariant functoriality for the diagonal morphism. In this way we obtain an intersection product on homology.

One can also work with Borel-Moore homology which is sometimes even nicer: under Poincaré duality the Borel-Moore homology corresponds to the usual cohomology, and in particular we get a unital ring, the unit being given by the fundamental class. And fundamental classes exist very generally in Borel-Moore homology, for example any irreducible complex analytic space (not necessarily smooth, not necessarily compact) has a fundamental class.

This is all easier than what Fulton does in his book. But Fulton obtains an intersection product on Chow groups, which carries more refined information.