# Topological definition of intersection multiplicities of algebraic varieties

Let $X$ be an algebraic variety over the field of complex numbers. In other words, $X$ is a reduced separable scheme of finite type over the field of complex numbers. Let $U$ and $V$ be irreducible subvarieities of X. Let $W$ be an irreducible component of $U ∩ V$. Suppose $W$ contains a closed non-singular point of $X$. In other words, the local ring of $X$ at $W$ is regular. Then dim $U$ + dim $V$ $≦$ dim $X$ + dim $W$ If the equality holds, one says that $U$ and $V$ intersect properly at $W$. In this case, a non negative integer called intersection multiplicity $i(U, V, W; X)$ is defined algebraically(see, for example, Serre's Local Algebra). I heard that this number can be defined by methods of algebraic topology. Is there any book which explains this in detail?

Edit I crossposted this on MathOverflow.

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Griffiths and Harris, Principles of Algebraic Geometry – user29743 Apr 25 '12 at 2:24
@countinghaus I have the book and I couldn't understand it well. It refers to archaic Lefschetz book. And I don't think their argument is rigorous enough. – Makoto Kato Apr 25 '12 at 2:32
hmm... I don't know another reference that says why the topological intersection multiplicities agree with the algebraic ones. If you just want a definition of the topological intersection product (without tying it back to algebraic geometry directly), you can get it in lots of places, but one of the best is Bott and Tu. You can check that it agrees with the algebraic definition by showing that it satisfies a short list of axioms which characterize the algebraic intersection product. – user29743 Apr 25 '12 at 2:38
Thanks, I'll check it. – Makoto Kato Apr 25 '12 at 2:41
I don't have it on hand to check, but I remember that Fulton's <i>Young Tableaux</i> has a good appendix on this. – David Speyer Apr 25 '12 at 5:26