Quantum cohomology for open varieties

Question

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.

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 $V$.

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.

Question: Is there such a thing for quantum cohomology?

I'm not sure functoriality of QC is established except for automorphism. Is that the only obstacle?

Answer 1

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.

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...

Answer 2

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)).

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 (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_ (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 http://arxiv.org/abs/1006.2501... though presumably it does a lot more.

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.