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I don't work on the Gromov-Witten theory, but I find that I need to study the following problem, which seems to be similar to the Gromov-Witten theory:

Let $X$ be a variety and $\alpha_{1}, \cdots, \alpha_{k}$ be some class on $X$, is there a morphism $f: (S,C_{1},\cdots,C_{k}) \rightarrow X$ from a surface $S$ with marked curves $C_{1}, \cdots, C_{k}$ such that $f_{i}(C_{i})$ lies in class $\alpha_{i}$?

In my situation the classes $\alpha_{i}$ form a loop, and curves $C_{i}$ are a circle of rational curves. I want to have some notion of "rationally simply connectedness" described by Gromov-Witten theory, similar to the description of "rationally connectedness" in Gromov-Witten invariants counting curves passing through two points. Also I want to have higher dimension analogues. Is it possible to do so?

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    $\begingroup$ Certainly there are cycle classes on moduli spaces of rational curves “counting” such surfaces. However, these are not (typically) symplectic invariants, nor do they have the other good properties of Groningen-Witten invariants. You can sometimes use Gromov-Witten invariants to deduce existence of rational surfaces (and this is a symplectic invariant). $\endgroup$
    – Jason Starr
    Commented Sep 24 at 8:06
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    $\begingroup$ There is "Moduli spaces M_{g,n}(W) for surfaces" by Valery Alexeev arxiv.org/pdf/alg-geom/9410003, which constructs a relevant moduli space. Perhaps forward searching for papers referring to this one will turn up something useful. $\endgroup$
    – Balazs
    Commented Sep 24 at 8:52
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    $\begingroup$ @Balazs There is no perfect obstruction theory, no virtual fundamental class, no recursion relations . . . $\endgroup$
    – Jason Starr
    Commented Sep 24 at 12:51

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Maybe this preprint by Y.Jiang is related: The virtual fundamental class for the moduli space of surfaces of general type.

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    $\begingroup$ Welcome new user. Rational surfaces are not of general type. $\endgroup$
    – Jason Starr
    Commented Sep 25 at 11:05

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