# Coarse moduli space versus Kuranishi family

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.

• 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. Jun 15 '19 at 11:36

The answer is yes: the germ of complex space $$(M_h, \, [X])$$ is analytically isomorphic to the quotient $$S/\mathrm{Aut}(X)$$.