# Chart for Deligne-Mumford Compactification

Suppose $g\ge 0$ and $n\ge 0$ are integers. We have the space $\overline{\mathcal M}_{g,n}$ of stable curves of arithmetic genus $g$ with $n$ marked points. The topology on this space is the one described in the paper "Compactness Results in SFT" (by Bourgeois et al) starting from page 811.

The notes "Holomorphic Curves in Symplectic and Contact Geometry" by Wendl, specifically Theorem 4.26, describe the orbifold structure of $\mathcal M_{g,n}$ (which is the space of non-singular curves of genus $g$ with $n$ marked points). This is done by means of a Banach manifold setup and the proof is an application of the implicit/inverse function theorem.

I have seen the following description (which I write here in the simplest case) of a neighborhood of a nodal curve in $\overline{\mathcal M}_{g,n}$ in https://arxiv.org/pdf/dg-ga/9608005.pdf. Assume that $(C_i,x_i)$ are stable curves of genus $g_i$ for $i=1,2$ with only one marked point. Then, the curve $C$ formed by joining $C_1$ and $C_2$ at $x_1$ and $x_2$ is stable. Choose holomorphic local isomorphisms $\varphi_i:U_i\to \mathbb D$ for $i=1,2$ such that $\varphi_i(x_i) = 0$ where $\mathbb D = \{z\in\mathbb C\;|\;|z|<1\}$.

Now, fix (non-empty) open subsets $K_i\subset C_i\setminus\overline U_i$, and choose charts for $\mathcal M_{g_i,1}$ around $(C_i,x_i)$ given by a family of almost complex structures $(J_i(z))_{z\in N_i}$ on the underlying smooth manifold $C_i$ parametrized by an open neighborhood $N_i$ of the origin in $\mathbb C^{(3g_i-3)+1}$ with the property that (i) at $0$, the almost complex structure is the original one, say $j_i$, and (ii) even at other points $z\in N_i$ the complex structure $J_i(z)$ coincides with the $j_i$ on the open set $C_i\setminus\overline K_i$. That this can be done is easy to see from the implicit function theorem argument (we have to use the principle of analytic continuation to show that "deformation can be localized").

Now, define the map $\psi:N_1\times N_2\times\mathbb D\to\overline{\mathcal M}_{g_1+g_2}$ as follows. $\psi(z_1,z_2,t) = \Sigma_{z_1,z_2,t}$, where $\Sigma_{z_1,z_2,t}$ is defined as follows. First take $C_i$ with the almost complex structures $J_i(z_i)$, and remove the closed set $\varphi_i^{-1}(\{|z|\le t\})$ from $C_i$. Now, we get $\Sigma_{z_1,z_2,t}$ be identifying the remaining annular parts of the $U_i$ via the holomorphic equivalence relation $p_1\in U_1\sim p_2\in U_2$ iff $\varphi_1(p_1)\varphi_2(p_2) = t$. When $t=0$, we interpret this as simply joining the two curves at the marked points. The claim is that $\psi$ (when restricted to a suitable neighborhood of $(0,0,0)$) maps in a finite to one manner (related to the action of the automorphism group of $C$) onto an open neighborhood of the nodal curve $C$.

The proof of this result in https://arxiv.org/pdf/dg-ga/9608005.pdf uses a lot of algebraic formalism from deformation theory which I'm not familiar with. I am interested in seeing a proof of this using some analytical methods (maybe have a suitable Fredholm analysis setup and combine with some gluing theorem).

I need to understand the analytic picture as I'm interested in learning about the gluing theorem for pseudoholomorphic curves in symplectic manifolds. I'd appreciate it very much if someone could outline an argument along these lines. I'm also ok with a reference or a sketch with the main ideas and in that case I'll try and work out the details myself.