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Manin stressed 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: "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.

Edit: Manin's and Smirnov's interesting computations and thoughts 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, part 2). BTW, has anyone the text by Kapranov which is mentioned in the paper above and in Hacking's introduction?

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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.

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"I'd like to know more" is the vaguest possible question :-) The best you can do is to watch the corresponding talk here - it is fantastic ! - and probably come back with more questions, since it is quite dense.

The background for the statement you are asking about is roughly the following (I am no expert and writing from memory - the following may contain nonsense and should not be taken literally):

Consider the moduli stacks $\overline{\mathcal{M}_{g,n}}$ of stable curves of genus g with n marked points. These are interconnected by a bunch of maps corresponding to glueing together stable curves, forgetting marked points, collapsing components to restore stability etc. (these are all operations with an intuitive geometric meaning and are wonderfully explained in Kock/Vainsencher's "Invitation to Quantum Cohomology").

If we map these into the category of motives, we can for each $n$ form their direct sum over $g$, thus getting a sequence of motives, and then assemble the above maps to form the structure maps of an operad, i.e. mainly "composition maps" $\overline{\mathcal{M}_{*,n_1}} \times \ldots \times \overline{\mathcal{M}_{*,n_k}} \rightarrow \overline{\mathcal{M}_{*,n_1 + \ldots + n_k}}$.

The word "motives" here means Chow motives and to make sense of this for stacks, one has to extend the definition of Chow groups to suitable stacks and then apply the usual constructions. This can be done in at least two different ways (Vistoli's and Toen's, see the talk for references).

Then the statement in the talk is that the motive of any smooth projective variety ("projective scheme" is a bit too bold because we are using Chow motives) is acted upon by this operad object in motives. To construct the action morphisms one uses cycles (and thus morphisms in motives) $I_{g,n}(V)$ corresponding to certain quantum cohomology classes, which are present in the quantum cohomology of any smooth projective variety...

Of course if you apply now any monoidal functor to all of this (e.g. a cohomology) you get an operad, and an object with an action from it, in the target category...

Edit (Thomas):

Here are Tom Coates' notes.

A try to summarize the seminar talks mentioned below: Only small quantum cohomology was discussed. The basic idea is to do the same as with normal motives, i.e. to map the cat. of varieties into something that looks like a Tannakian category. Whereas usually, one seeks to classify and identify motives by their corresponding representations of Gal(Q), here on looks for representations of a bigger "quantum Galois group of Q" found within known representations coming from geometry. That "quantum Galois group" was not described, maybe this relates to it?

Golyshev mentioned as motivating analogy an article by Deligne, where D. proves the Weil-conjectures for K3-surfaces by embedding motives from them in a product of motives from abelian varieties, for whom the Weil-conj.s were known to be true. That embedding of motives was done by looking how the representations fit together, and by identifying whose rep.s come from abelian varieties. In Golyshev's analogy, even dim. quadrics should play the role of K3-surfaces, the orth. group the role of a construction by Kuga-Satake in Deligne's paper on the repr.s of ab. var.s

A search method for such representations which would led to "quantum motives" is made with help of mirror symmetry. That idea seems to go back to Golyshev.

Coates' method seems to be: The q.-cohomology of Fanos is known, that of toric varieties is not; On the other hand, the Laurent polynomials of toric varieties are known, but that of Fanos are not. Mirror symmetry connects q.-cohomology with such polynomials and Golyshev has a conjecture conc. the Laurent-poly.s of Fanos. Coates' computations try to bridge between Fanos and toric varieties by deformation theory and look if polynomials result which look like coming from pieces of Fanos, which then could be "quantum motives".

Maniel's idea is to look at the quantum cohomology of Grassmanians etc., (if I understood him correctly) producing other interesting representations through a quantum Satake correspondence. Unfortunately he mentioned that with the quantum Satake so shortly at the beginning of his talk, that I am unsure if my idea on it's use is correct.

A very fascinating talk was by Gorbounov, he plays around with Landau-Ginzberg potentials and finds quantum cohomology as special case of equivariant coh. However that may be, as far as I know, Landau-Ginzberg potentials are only used in Grassmanians etc., q.-coh. is not restricted to them.

One talk I understood nothing from was by Katzarkov, probably similar to his Oberwolfach talk earlier this year. The connection to quantum motives is unclear to me.

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Whoops, I didn't see that the question was that old - you are probably way beyond that now... –  Peter Arndt Sep 5 '10 at 12:37
    
Actually, Manin's talk had made me read into that theme a bit, but I thought then, such a supergeneral question could provoke good reactions, similar to letting now an old question unchanged ;)), being still busy with Manin's book on that theme (sketching an amazing speed of development in the first few years after the theme started) and wondering what the coming talks tell, if they are accessible... BTW, that is interesting too: math.ntu.edu.tw/~linhw/papers/ICCMProceeding-HuiWenLin.pdf –  Thomas Riepe Sep 5 '10 at 13:34
    
That seminar sounds nice indeed. If any notes or videos appear online I would be highly interested. If not, I will in ten days post the question "What are quantum motives?" :-) –  Peter Arndt Sep 5 '10 at 16:51
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About monodromy and quantum cohomology: while everybody knows that Gromov-Witten invariants are invariant under deformations, the immediate consequence that they are invariant under the monodromy action for any smooth family of varieties is sometimes forgotten. In other words, they are invariant under the mapping class group. This can be extremely useful in computations when the mapping class group is big.

To elaborate a bit: when we are saying "GW invariants are invariant under small deformations", then we are implicitly already using the Gauss-Manin connection: If $X \to T$ is a family, say over the disc, then it doesn't make sense to identify Gromov-Witten invariants of $X_{t_1}$ and $X_{t_2}$ unless you know which GW-invariants to compare; in other words, given cohomology classes $\gamma_1, \dots, \gamma_n$ and a homology class $\beta$ on $X_{t_1}$, we need to find corresponding classes on $X_{t_2}$. Well, fortunately, this is not a problem, as $X_{t_1} \cong X_{t_2}$ as smooth manifolds, and thus $H^*(X_{t_1}) \cong H^*(X_{t_2})$. This identification is nothing but the Gauss-Manin connection. In particular, when the bases $T$ is more complicated, you need to choose a path from $t_1$ to $t_2$ to obtain the identification; and it will depend on the homotopy class in the path. In particular, when $T$ is not simply-connected, we get a representation of $\pi_1(T)$ on $H^*(X_0)$, i.e. the monodromy action. Since it is pieced together out of identifications $H^*(X_0) \cong H^*(X_{t_1}) \cong H^*(X_{t_2}) \cong \dots$ as above, GW-invariants are invariant under this group action:

$ \langle \gamma_1, \dots, \gamma_n \rangle_{\beta}^{g,n} = \langle \Phi(\gamma_1), \dots, \Phi(\gamma_n) \rangle_{\Phi(\beta)}^{g,n} $

for any $\Phi$ in the image of $\pi_1(T) \to \mathrm{Aut} H^*(X, \mathbb{Q})$. (The mapping class group is basically the biggest possible group $\pi_1(T)$ you can obtain this way, i.e. the fundamental group of the moduli space of varieties diffeomorphic to $X$.)

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Can you elaborate? I don't quite understand what you're saying. –  Kevin H. Lin Sep 6 '10 at 7:00
    
I suppose this was way more details than you were asking for, but I hope it is clearer now. –  Arend Bayer Sep 7 '10 at 11:30
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