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For which manifolds or varieties is quantum cohomology known to converge? Are there any manifolds for which quantum cohomology is known to not converge? I seem to have the impression that quantum cohomology is known to converge for Fano manifolds or maybe toric Fano manifolds, but I don't know if this is actually true.

By "convergence" I mean convergence of, e.g., the genus 0 Gromov-Witten potential.

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    $\begingroup$ Maybe you have seen (frist page)arxiv.org/PS_cache/math/pdf/0506/0506236v4.pdf Overwise, I guess you could ask Tom Coates. Once he told me he knows how to prove convergence for Fanos, but maybe he was joking :) What is true, for some reason people tend to leave this question aside... $\endgroup$
    – Dmitri Panov
    Commented Dec 26, 2009 at 12:58
  • $\begingroup$ Oh yeah, thanks, I've seen that paper before, but I had forgotten about it. That's probably where I got my impression about toric Fanos... I should have just googled :), it's the second thing (after this very page!) that comes up when you google "convergence of quantum cohomology". Theorem 1.3 states that big quantum cohomology for smooth projective toric varieties converges. The first page also seems to claim that convergence of small QH is trivial for the case of positive c1, and convergence of big QH is trivial for negative c1, but I'll have to think about that... $\endgroup$
    – Kevin H. Lin
    Commented Dec 26, 2009 at 13:46
  • $\begingroup$ Ok, so now I have a fairly satisfactory answer. I'm not sure what I should do about the bounty now, since that answer is living in a comment, and I can't award the bounty to a comment. On the other hand I guess I'll wait a bit to see if anybody else has anything interesting to add. $\endgroup$
    – Kevin H. Lin
    Commented Dec 26, 2009 at 14:01
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    $\begingroup$ You could ask Dmitri to submit this comment as an answer, and award him for that. $\endgroup$
    – GMRA
    Commented Dec 27, 2009 at 2:55

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There are many more examples of convergent quantum cohomology generating functions where the proof follows from mirror symmetry. MS for, e.g., the quintic says the QC generating function is the power series expansion of a hypergeometric function, and thus it converges.

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See Dmitri's comment.

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