This question is related with my previous one Quantum cohomology rings as invariants, but now, I want to ask a more concrete thing. If $X$ and $Y$ are Poisson varieties which are isomorphic (as a Poisson varieties) then, Are their quantum cohomology rings isomorphic?
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$\begingroup$ Cat, what class of mathematical objects have quantum cohomology? Which objects have isomorphic quantum cohomology rings by definition? $\endgroup$– Tim PerutzCommented Nov 30, 2010 at 14:39
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Thanks Tim. Actually, I was thinking in $X$ and $Y$ as symplectic varieties, I mean, as two Poisson varieties which are symplectic. In this case if the varieties are isomorphic as symplectic varieties their quantum cohomology rings are isomorphic. My question was really stupid. I apologise for that. Please, close it.
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$\begingroup$ Be careful about what symplectic (or Poisson) means! There are differentiable and algebraic versions of these notions. QH is an invariant of compact (differentiable) symplectic manifolds. A non-singular projective variety over $\mathbb{C}$ (in the analytic topology) is canonically symplectic, hence has quantum cohomology. One can calculate this algebraically. $\endgroup$ Commented Nov 30, 2010 at 20:46
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$\begingroup$ ...If "symplectic variety" means an alg variety with an algebraic symplectic form, then you're missing important info for QH (the projective embedding, which determines the weights to count curves) but also have irrelevant info (the alg symp form). $\endgroup$ Commented Nov 30, 2010 at 20:46