Quantum cohomology for open varieties - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-23T12:17:58Z http://mathoverflow.net/feeds/question/60742 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/60742/quantum-cohomology-for-open-varieties Quantum cohomology for open varieties YBL 2011-04-05T21:04:27Z 2011-04-06T02:08:14Z <p>Hi, I know very little about the quantum cohomology (QC for short). I only got interested in the subject as the genus zero part may be relevant to a problem I'm working on. So I hope my question makes sense. </p> <p>I understand QC defines a structure of an algebra over the operad $H^*(\bar{M}_{g,n})$ on the cohomology $H^*(V)$ of any smooth projective complex variety $V$. </p> <p>In Hodge theory, the yoga of weight filtration extends enriched structure on the cohomology from smooth projective varieties to smooth varieties. This is done by using resolution of singularities to represent a quasi-projective smooth variety $X$ as the complement $\bar{X} \setminus D$ of a normal crossing divisor in a projective variety. Then one can compute the cohomology $H^* (X)$ in terms of the cohomology $H^* (D_q)$ of the closed stata using a spectral sequence. </p> <p>Question: Is there such a thing for quantum cohomology? </p> <p>I'm not sure functoriality of QC is established except for automorphism. Is that the only obstacle? </p> http://mathoverflow.net/questions/60742/quantum-cohomology-for-open-varieties/60759#60759 Answer by Alexander Braverman for Quantum cohomology for open varieties Alexander Braverman 2011-04-06T01:41:48Z 2011-04-06T01:57:21Z <p>I think that if you take an affine variety, all of its Gromov-Witten invariants of degree $\neq 0$ are $0$ in any sense. So QC for affine varieties should coincide with ordinary cohomology.</p> <p>The following point of view might be useful: you might (and in fact, should) think about small quantum cohomology as some kind of Floer cohomology of the loop space of $X$. If you cover $X$ by subsets, then the loop space is NOT covered by the corresponding loop spaces, so your idea doesn't seem right to me...</p> http://mathoverflow.net/questions/60742/quantum-cohomology-for-open-varieties/60760#60760 Answer by Daniel Pomerleano for Quantum cohomology for open varieties Daniel Pomerleano 2011-04-06T02:08:14Z 2011-04-06T02:08:14Z <p>I'm just starting to learn this stuff myself, but here is a possible way to think about Alexander's answer. Given a compact smooth variety X and a divisor D we have the Fukaya category of X and the Fukaya category of X-D. If you look at Seidel's (2002 ?) ICM address, it says that under good circustances we can think of the Fukaya category of X as a deformation of the fukaya category of the open manifold U = X-D (basically the question is under what circumstances there are enough objects of Fuk(X-D) to generate Fuk(X)). </p> <p>Given an $A(\infty)$ category like the Fukaya category, we can take something called Hochschild cohomology, HH*. If U is say the cotangent bundle of a manifold M, HH*(Fuk(U)) is isomorphic to H_*(LM), but in general it is something known as symplectic homology. HH <em>(Fuk(X)) is (conjecturally in general) quantum cohomology. In principal, if you somehow understand that deformation class really well, there would be some sort of spectral sequence from SH_</em> (U) to quantum cohomology, but I can't think of a situation where it would be easier to compute this than quantum cohomology itself. I think this paper of Eliashberg and Polterovich might be in the spirit of such a computation from an Symplectic field theory point of view <a href="http://arxiv.org/abs/1006.2501" rel="nofollow">http://arxiv.org/abs/1006.2501</a>... though presumably it does a lot more.</p> <p>A different idea that is often used is to work the other way around and try to use relative Gromov Witten theory of a bigger manifold e.g. projective space to compute Gromov Witten invariants of a hypersurface. For this you can see work by Andreas Gathmann.</p>