We will work over complex number field $\mathbb{C}$. Let $\mathscr{M}\_h$ be the moduli functor for canonically polarized manifolds with $h$ the Hilbert polynomial. Let us denote by $M\_h$ the coarse moduli space of $\mathcal{M}\_h$. For any $X\in \mathscr{M}\_h({\rm Spec}(\mathbb{C}))$, we know that its Kuranishi family $f:\mathscr{X}\to S$ exists, with $f^{-1}(s\_0)\simeq X$. After shrinking $S$, we may assume that $f:\mathscr{X}\to S\in \mathscr{M}\_h(S)$. Since $H^0(X,T\_X)=0$, the Kuranishi family $f:\mathscr{X}\to S$ is universal. Moreover, the finite automorphism group $\mathrm{Aut}(X)$ acts on $S$ with $s\_0$ the fixed point due to the definition of Kuranishi family. 


My question is: can $S/\mathrm{Aut}(X)$ be seen as a neighborhood of $[X]\in M\_h$? In other words, $M\_h$ can be obtained by glueing together the quotient of Kuranishi spaces. 



Coarse moduli space versus Kuranishi family
The answer is *yes*: the germ of complex space $(M\_h, \, [X])$ is analytically isomorphic to the quotient $S/\mathrm{Aut}(X)$. 


This is true not only for moduli spaces of hyperbolic curves, but also in the (much more difficult) context of Gieseker moduli space of (canonical models of) surfaces of general type, see Remark 3.7 in 


F. Catanese, [*A superficial working guide to deformations and moduli*](http://www.arxiv.org/abs/1106.1368), Farkas, Gavril (ed.) et al., Handbook of moduli. Volume I. Somerville, MA: International Press; Beijing: Higher Education Press (ISBN 978-1-57146-257-2/pbk; 978-1-57146-265-7/set). Advanced Lectures in Mathematics (ALM) **24**, 161-215 (2013). [ZBL1322.14002](https://zbmath.org/?q=an:1322.14002).


