In Fulton's Intersection Theory, he develops the notation $\int_X$ for the degree homomorphism from $A^*(X)$ to $\mathbb{Z}$, and I was wondering if there was a reason for the notation. Is this in any sense a kind of integration?

  • $\begingroup$ The way he defines this there's really only something going on for $A_0$, right? $\endgroup$ – Hoot Jul 12 '15 at 5:37
  • $\begingroup$ Yeah. For all higher (or lower, depending on your point of view) cycles it's just $0$. But the fact that he bothers to make this extension in the first place suggested to me there was something more general going on, of which this was just an instance. $\endgroup$ – Peter Xu Jul 12 '15 at 5:39
  • 1
    $\begingroup$ Relevant meta discussion, by the way. $\endgroup$ – HDE 226868 Jul 12 '15 at 13:32
  • 3
    $\begingroup$ For those still interested in the transcript of my conversation with Peter surrounding this question, see here: etreseul.wordpress.com/2015/07/12/… $\endgroup$ – user149792 Jul 12 '15 at 20:40

Oh, it corresponds to integration of top-level differential forms over the analytic space when $A^*$ is identified with the (even) cohomology ring.

  • $\begingroup$ There is some duality at work here isn't there? $\endgroup$ – Hoot Jul 12 '15 at 18:08
  • $\begingroup$ A cycle $\alpha$ is associated to the homology class associated to its (linear combination of) submanifolds in the analytification of $X$, which is then associated to a cohomology class via Poincare duality. Is this what you mean? $\endgroup$ – Peter Xu Jul 12 '15 at 18:54

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.