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 non-variety things.

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. There is also a K-theoretic version of quantum cohomology, appropriately named quantum K-theory. This structure comes from a K-theoretic version of GW invariants. In the algebraic context, there is something called a motive, which is supposed to be in some sense "the universal cohomology theory" -- here cohomology theory does not mean Eilenberg-Steenrod cohomology theory, but rather (I think) Weil cohomology theory. I don't know anything about motives, but I guess Manin is saying that just as you can do cohomological (and Chow) and K-theoretic versions of GW invariants and thus quantum cohomology and quantum K-theory, you can also do the same or analogous thing for 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.

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.

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 non-variety things.

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. There is also a K-theoretic version of quantum cohomology, appropriately named quantum K-theory. This structure comes from a K-theoretic version of GW invariants. In the algebraic context, there is something called a motive, which is supposed to be in some sense "the universal cohomology theory". I don't know anything about motives, but I guess Manin is saying that just as you can do cohomological and K-theoretic versions of GW invariants and thus quantum cohomology and quantum K-theory, you can also do the same or analogous thing for 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.

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 non-variety things.

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. There is also a K-theoretic version of quantum cohomology, appropriately named quantum K-theory. This structure comes from a K-theoretic version of GW invariants. In the algebraic context, there is something called a motive, which is supposed to be in some sense "the universal cohomology theory" -- here cohomology theory does not mean Eilenberg-Steenrod cohomology theory, but rather (I think) Weil cohomology theory. I don't know anything about motives, but I guess Manin is saying that just as you can do cohomological (and Chow) and K-theoretic versions of GW invariants and thus quantum cohomology and quantum K-theory, you can also do the same or analogous thing for 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.