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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.

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  • $\begingroup$ Let me mention that the paper by Fujiki "Coarse moduli space for polarized compact Kähler manifolds" ems-ph.org/JOURNALS/… established a more general result than I expected in the question. $\endgroup$ Jun 15 '19 at 11:36
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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, 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.

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  • $\begingroup$ Thanks for your quick answer! I took a quick look at Catanese's paper, and I think he mainly studied the case for surfaces. Does your answer hold for moduli of higher dimensional canonically polarized manifolds? $\endgroup$ Jun 11 '19 at 13:35
  • $\begingroup$ I guess so, but one must take "moduli space" in the right sense (canonical models, usually). Try to have a look at some of J. Kollar's papers, for instance Moduli of varieties of general type. $\endgroup$ Jun 11 '19 at 13:41

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