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? 



Quantum cohomology for open varieties
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...


