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So, a basic set-up in modern enumerative geometry is that you have some object $X$ (say, a "nice" stack, for whatever definition of "nice" you need) and then you want to "count" the curve of genus $g$ intersecting a bunch of cohomology classes and of degree $\beta$, so you look at $\bar{\mathcal{M}}_{g,n}(X,\beta)$, then pull back the classes, intersect, and get a Gromov-witten Invariant. Famously, this gives the Kontsevich formula counting rational curves in the projective plane passing through $3d-1$ points. And though GW-invariants can be negative and rational, there are nice cases where they do count something legitimate, such as the genus $0$, $n\geq 3$ case into a homogeneous space.

So, enough background, here's my question (and this is largely idle curiosity, so no specific motivation): can we do this in higher dimensions? For instance, given a (smooth?) variety $V$ and marking a bunch of subvarieties $W_1,\ldots,W_n$ (maybe restricting them to points?) can we form $\bar{\mathcal{M}}_{V,(W_1,\ldots,W_n)}(X,\beta)$ a moduli space of stable mappings of varieties deformation equivalent to the one we started with into our space, represented by a given cohomology class $\beta$ and with $W_1,\ldots,W_n$ intersecting some cohomology classes, so that we can get something that can be called higher dimensional Gromov-Witten invariants? If this has been studied, under what conditions does it actually count subvarieties? For instance, if $X$ is $\mathbb{P}^N$ and $V=\mathbb{P}^2$, and maybe if we loosen things to just needing rational maps, could we use something like this to count rational surfaces, satisfying some incidence conditions?

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The ordinary GW theory depends on two things: that the moduli space of stable curves/maps is proper, and that it has a perfect obstruction theory. Maybe a good first test would be to check whether the higher dimensional generalization of the former is still proper -- or is it obvious somehow? – Kevin H. Lin Dec 22 '09 at 18:02
I don't know. This really arose as idle curiosity on my part while reading a bunch of papers by Pandharipande and his collaborators about ordinary GW theory. I'd love it if people who knew about properness posted here. – Charles Siegel Dec 22 '09 at 18:13
Actually, I just realized that my comment was perhaps poorly phrased, and that the real first question should be: how can we define the higher dimensional generalizations in the first place? Are there nice moduli spaces of surfaces or $n$-folds or whatever? Are there nice compactifications thereof? To mimic the structure of ordinary GW theory, the choice of compactification should maybe be such that it still parameterizes things that are gotten by gluing smooth things together along marked points...? I don't know, I'm just making things up, really. – Kevin H. Lin Dec 22 '09 at 19:07
Kevin: that's always a good thing to do first, just make things up and hope it works. I don't know much about moduli spaces of varieties in dimension $\geq 2$, other than that you can deform the complex structure to not be algebraic anymore, at least for K3 Surfaces and complex tori. Perhaps the right first thought should be to try something like "stable abelian varieties" because there are good, well known compactifications of $\mathcal{A}_g$? – Charles Siegel Dec 23 '09 at 2:35

I know that you are thinking firmly about the integrable world, but I thought it worth adding that for symplectic manifolds, there is no obvious generalisation of Gromov-Witten theory to higher dimensional subvarieties. This is because to define "holomorphic" you use a non-integrable almost complex structure and non-integrability means that there are no higher dimensional holomorphic objects. The fact that there are holomorphic curves can be thought of as an instance of the fact that all almost complex structures over 2-manifolds are automatically integrable. (E.g., since there are no (2,0)-forms, the space where the Nijenhuis tensor should live is zero.)

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Ok, so if anyone were to try this, they would essentially have to do it algebraically, because the symplectic case just doesn't make sense, at least, not in general? Though in the case of an integrable almost complex structure, it might be made to work out? – Charles Siegel Dec 20 '09 at 13:35
Yes, that's right. My understanding is that if you tried on a symplectic manifold with a generic almost complex structure, the moduli spaces for higher dimensional pseudoholomorphic submanifolds would just be empty. Any enumerative problems of this sort are purely part of holomorphic geometry. Which is, of course, no bad thing in itself! – Joel Fine Dec 20 '09 at 14:57

It seems pretty much impossible that a virtual fundamental class for a moduli space of maps from surfaces to a variety can be constructed with current techniques. In the case of curves, deformations of a map are controlled by $H^0(C, f^* T_X)$, and obstructions by $H^1(C, f^* T_X)$. These two spaces vary nicely over the moduli space, in the sense that there is a two-term complex on the moduli space whose fiber at the point [C, f] has these cohomologies. (It is obtained by pulling back the tangent bundle to the universal curve, then taking its derived push-forward along the projection to the moduli space.) If you do the same with surfaces, you end up with a 3-term complex. In that case, Behrend-Fantechi no longer help you to construct a virtual fundamental class.

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Alexeev's and Knutson's moduli space of branch varieties is kind of like a higher dimensional version of the moduli space of stable maps. (Although you should be warned that the moduli space of stable maps is not a special case of the moduli space of branch varieties. In the moduli space of branch varieties, there are never any maps with positive dimensional fibers.)

I don't know of anyone whose done serious enumerative work along these lines. I'm looking forward to reading the other answers.

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That does look interesting. My biggest thought is that maybe nothing's been done in this direction because it's hard enough to get decent results for curves, trying even for surfaces might be too much. – Charles Siegel Dec 20 '09 at 1:40
Charles: Well, with the proper point of view, it's not that hard to get some decent results for curves. For example, counting rational curves in $\mathbb{P}^2$ reduces to associativity of quantum cohomology, which is by now a very "standard" fact. It is hard to argue against the claim that it is the physics(-inspired) structures -- like quantum cohomology -- which give GW theory its richness. Perhaps one can speculate that to find a good notion of higher dimensional GW theory, we could try to look for a good physical interpretation... – Kevin H. Lin Dec 22 '09 at 17:54
David: Thanks for posting that paper, it looks very interesting. You say you don't know of anyone who has done serious enumerative work along those lines; how about anyone who has done not-so-serious enumerative work? ;-) – Kevin H. Lin Dec 22 '09 at 19:26

The moduli space of stable maps to a point is the Deligne-Mumford moduli space of curves. The moduli theory of higher-dimensional varieties is extremely complicated: though much has been done, even for surfaces there are many possible stability conditions. So even the "higher Gromov-Witten theory of a point" would be very challenging to define.

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The moduli space has been constructed:

However, I don't think anyone proved anything more than that. In particular, I don't know if the existence of a virtual fundamental class is known for it.

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I don't know if this has any bearing on the question, but R. Vakil has shown that moduli spaces of surfaces with very ample canonical bundle can have "arbitrarily bad" singularities.

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In theory a statement of this sort could be used that a perfect obstruction theory cannot exist - but you would need effective bounds. ( like "I know this moduli space has expected dimension 10, but whenever we embed our local singularity of the moduli space into a smooth variety of dimension 10+d, then we need at least d+1 equations to define the embedding.") – Arend Bayer Jul 24 '10 at 19:15

As ABayer said since the domain is no longer a curve, there are higher obsturction and one does not know how to define the virtual cycle. By the way, one can consider a related problem: try to count special lagrangians (in general calibrated submanifolds). It turns out that the deformation of calibrated submanifolds is unobsturcted. But one meets another problem to define a invariant, i.e the compactification issue, one usually does not know how to compactify the moduli space of calibrated submanifolds.

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