Suppose $(X,\omega,J)$ is a compact Kähler manifold, and $\beta\in H\_2(X,\mathbb Z)$ is given. Then, we can form the space $\overline{\mathcal M}:=\overline{\mathcal M}\_{0,0}(X,\beta)$ of stable maps $u:C\to X$ with $C$ a nodal curve of genus 0 and $u\_\*[C]=\beta$. Let $\mathcal M\subseteq\overline{\mathcal M}$ be the locus where the domain is nonsingular.


I know how to show that $\mathcal M$ is a space with a smooth ($C^\infty$) Kuranishi atlas (i.e., roughly, it locally has the structure of the zero set of a smooth vector valued function on a smooth finite dimensional manifold modulo a finite group). Near the points of $\overline{\mathcal M}\setminus\mathcal M$, I don't know how to put a smooth structure (instead, what I know amounts to only a topological implicit atlas, in the sense of <https://arxiv.org/abs/1309.2370>).


My question is the following: since I have assumed $X$ actually Kähler (and not simply symplectic with a compatible almost complex structure), can we show that $\mathcal M$ (or even better, $\overline{\mathcal M}$) has a complex analytic Kuranishi atlas? I would prefer approaches which stay within the realm of differential geometry (though for the case of projective $X$, I would be interested in hearing about algebro-geometric approaches as well). Any references would also be appreciated.



Complex Analytic Structure on Moduli Space of Stable Maps
There's also a really beautiful approach by Salamon-Robbin-Ruan for integrable Js in their paper "The moduli space of regular stable maps"


<https://people.math.ethz.ch/~salamon/PREPRINTS/smUW.pdf>


building on the earlier paper by Salamon-Robbin "Construction of the Deligne-Mumford orbifold":


<https://people.math.ethz.ch/~salamon/PREPRINTS/dmETH.pdf>


They gives a smooth structure that extends across the boundary in an extremely natural way (I don't remember if they explicitly construct an analytic structure, but because they are working in the case of integrable J, I think it should be possible to extract it from their proof).


The idea is the following.


Consider the space of holomorphic maps $S^2\to X$. Since a holomorphic map on the sphere is determined by its values on the equator, you can think of this as a submanifold of the space of maps from the circle to $X$. In fact, it's precisely the intersection of the (infinite-dimensional) subspace of maps $S^1\to X$ which extend over the northern hemisphere with the (infinite-dimensional) subspace of maps which extend over the southern hemisphere (this is the kind of thing people mean when they say that Floer theory is the theory of intersections of semi-infinite cycles).


Now when your holomorphic map develops a bubble, this bubble will (generically) avoid the equator. So if you think of your moduli space as a space of maps $S^1\to X$ then you don't even notice bubbling.


I think their original hope was to use this for nonintegrable Js as well, and overcome all issues with smooth structures at the boundary of moduli spaces. The problem, I believe, is that they need to use Fourier theory and Hardy spaces to set things up in a way they can handle analytically. Why does Fourier theory show up? A simple heuristic: a map from the circle to $\mathbb{C}$ can be thought of as a power series in $z$ and $z^{-1}$ infinite in both directions; those which extend over the unit disc are those where the negative powers of $z$ are absent; those which extend over the north hemisphere of the Riemann sphere are those where the positive powers of $z$ are absent; the only holomorphic maps from $S^2$ to $\mathbb{C}$ are therefore the constants. The coefficients of $z^n$ can be thought of as Fourier coefficients. This kind of description of holomorphic maps as power series only makes sense in the integrable setting.


