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This is a very basic question about the definition of Moduli space of maps. My reason for asking this question is because I haven't actually seen this definition explicitly given anywhere, and hence I am wondering if I am mistaken for some reason.

$\textbf{Question $1$:}$ Let $\Sigma_g$ be a smooth genus $g$ surface and $j \in \mathcal{M}_{g,0} $ a complex structure on $\Sigma_g$. I am assuming that I ``understand'' $\mathcal{M}_{g,0}$, the space of complex structures on $\Sigma_g$ as a topological space, not just as a set. I want to know if the following is a correct definition of $\mathcal{M}_{g,k}(\mathbb{P}^2, d) $, the moduli space of genus $g$ degree $d$ curves in $\mathbb{P}^2$, with $k$ parked points:

$$ \mathcal{M}_{g,k}(\mathbb{P}^2, d):= \{ (u; y_1, \ldots, y_k; j) \in \mathcal{C}^{\infty}_d(\Sigma_g, \mathbb{P}^2) \times ((\Sigma_g)^k-\Delta )\times \mathcal{M}_{g,0}: \overline{\partial}_{j} u =0\}/\sim, $$

where $\mathcal{C}^{\infty}_d(\Sigma_g, \mathbb{P}^2)$ denotes smooth degree $d$ maps, $\Delta$ denotes the union of all diagonals in $(\Sigma_g)^k$ and $\sim $ is the equivalence relation that if $\sigma$ is an Automorphism of $(\Sigma_g, j)$, then $$ (u, y_1, \ldots, y_k; j) \sim (u\circ \sigma, \sigma^{-1}(y_1), \ldots, \sigma^{-1}(y_k);j). $$

Furthermore, is it true that $\mathcal{M}_{g,k}^*(\mathbb{P}^2, d)$, the subspace of curves that are somewhere injective (non multiply covered) is a smooth manifold of the expected dimension if the "Linearization" of $\overline{\partial}_{j} u$ is surjective?

$\textbf{Remark:}$ These statements are all true if $g=0$. There is only one complex structure on the domain. This is the way moduli spaces are defined in Symplectic Geometry (in particular this is how they are defined in McDuff and Salamon's book). I basically want to know if there is a similar description for the moduli space of $(J, j) $ holomorphic maps, where we are allowed to vary the complex structure on the domain. If we keep the complex structure on the domain to be fixed, then there is conceptually nothing different from genus $0$ and higher genus $g$.

$\textbf{Question $2$:}$ I also want to know if the compactification of this moduli space is the correct space to consider while defining higher genus Gromov Witten Invariants (i.e. are higher genus Gromov Witten invariants intersection of some chomology classes on $\bar{\mathcal{M}}_{g,k}(\mathbb{P}^2, d)$) where $\mathcal{M}_{g,k}(\mathbb{P}^2, d)$ is how I have defined it to be?

$\textbf{Question $3$}$ Is it correct to say that naively one expects higher genus Gromov Witten Invariants to be counts of genus $g$ curves with a $\textit{variable}$ complex structure on the domain? To clarify this point, if we are looking at Enumerative Geometry of higher genus curves there are two inequivalent questions one can ask: how many genus $g$ curves are there with a $\textit{fixed complex}$ structure or how many curves are there with a $\textit{variable}$ complex structure; aside from genus $0$ these two questions are completely different. Is it correct to say that naively one expects Gromov Witten invariants to be the answer to the second question? Of course in practice Gromov Witten Invariants are almost never going to be enumerative for higher genus curves, but that is not my question here.

$\textbf{Added Later:} $ Regarding Question $1$, let me make one further remark. The expected dimension of the moduli space is $$ <c_1(TX), ~[\beta]> + (\mathrm{dim}(X)-3)(1-g) +n $$ where I have considered the more general case of maps of degree $\beta $ into a manifold $X$. Is it possible to interpret the expected dimension as the index of the Linearization of some section?

My main source of confusion is how do I think about the moduli space of curves (or a perturbed moduli space of curves if that makes the question more meaningful, i.e. curves that satisfy the equation $\overline{\partial} u = \nu $, where $\nu$ is a perturbation).

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  • $\begingroup$ I suggest looking up the definition of a cohomological field theory in the sense of Kontsevich and Manin. This is "the standard data" that one should be able to extract out of Gromov-Witten invariants using whatever transversality package one envsions using. $\endgroup$ Jun 12, 2015 at 11:35
  • $\begingroup$ @Daniel: I have looked Kontsevich-Mannin's paper; my main confusion is how does one define the Moduli space of curves as the zero set of the section of some infinite dimensional bundle (i.e the section basically being d bar), since the d bar keeps changing with the complex structure $j$ on the domain. Basically, I am looking for some interpretation that is in spirit similar to the one given in McDuff and Salamon. $\endgroup$
    – Ritwik
    Jun 12, 2015 at 12:04
  • $\begingroup$ These are all contained in Ruan-Tian's original paper. $\endgroup$
    – Guangbo Xu
    Aug 28, 2015 at 1:47

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