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?
Best Answer
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.)