Manin [stressed](http://www.ihes.fr/document?id=1704&id_attribute=48 "Manin talk") that every projective scheme should have a quantum-cohomology structure. I'd like to know more about that. And since the varieties considered in texts about monodromy resp. vanishing cycles which I have read are projective, I am curious about the behaviour of quantum cohomology under monodromy.


Edit: A coming [seminar](http://www.mpim-bonn.mpg.de/Events/This+Year+and+Prospect/Seminar+Quantum+Motives/ "link"): "Quantum motives: realizations, detection, applications", incl. a lecture "Quantum motives: review (of) the classical idea of how to linearize algebraic geometry with an eye to utilizing it in the quantum setup." and a minicourse "Geometric Langlands and quantum motives: a link". 


Edit: The slides on Manin's talk on the concept of classical motives and it's relation to quantum cohomology are [here](http://pdfcast.org/pdf/manin-s-2009-ihes-talk-quantum-motives "some free pdf-sharing site"). 


Edit: Manin's and Smirnov's [interesting computations and thoughts](http://arxiv.org/abs/1107.4915 "link") on e.g. a "membrane quantum cohomology" (**+** russian **videos** of a talk in june 2011 on the program "... elucidation of this "self-referentiality" of quantum cohomology has just begun. The talk tries to outline the contours of this huge program and the first steps of it.": [part 1](http://narod.ru/disk/17105482001/Globus-Manin-p1.mp4.html "download-site"), [part 2](http://narod.ru/disk/17109473001/Globus-Manin-p2.mp4.html "download-site")). BTW, has anyone the text by Kapranov which is mentioned in the paper above and in [Hacking's introduction](http://arxiv.org/abs/math/0509567 "arxiv")? 



ubiquitous quantum cohomology
I'm not sure what you mean by "every projective scheme should have a quantum cohomology structure". In the talk abstract that you link to, it does not say "projective scheme" but "smooth projective variety". I don't know whether the theory generalizes to non-smooth things or to things that are not varieties.


Quantum cohomology is a deformation of ordinary cohomology (or Chow ring if you like) of a smooth projective variety (or compact symplectic manifold). This structure comes from (genus 0) Gromov-Witten invariants. GW invariants are constructed using the ordinary cohomology of your variety/manifold and the ordinary cohomology of moduli spaces of stable maps and stable curves. I mostly work over $\mathbb{C}$ so I don't know too much about what I'm about to say, but if you're not working over $\mathbb{C}$, then ordinary cohomology doesn't make sense, but instead you can still work with things like $\ell$-adic cohomology or crystalline cohomology. This is what "motives" refers to. I guess Manin is saying that just as you can do cohomological (and Chow) Gromov-Witten invariants and quantum cohomology, you can also do the analogous things for motives. I suppose the resulting things would be called "motivic Gromov-Witten invariants" and "quantum motives".


I'm not sure whether it makes sense to ask about the behavior of quantum cohomology under monodromy. As I understand it, monodromy refers to using a connection (Gauss-Manin connection) to parallel transport (co)homology classes. You can view quantum cohomology as being simply ordinary cohomology except with coefficients in a Novikov ring and with a deformed cup product. Viewed as such, the monodromy of quantum cohomology should be the same as the monodromy of ordinary cohomology, because "quantum cohomology classes" are no different from ordinary cohomology classes.


