There is the Proposition 6.4.3 in Donaldson-Kronheimer as follows:
Proposition (6.4.3)
(i) There is a holomorphic map $\psi$ from a neighborhood of $0$ in $H^1(\operatorname{End} \mathscr{E})$ to $H^2(\operatorname{End}_0 \mathscr{E})$, with $\psi$ and its derivative both vanishing at $0$, and a versal deformation of $\mathscr{E}$ parametrized by $Y$ where $Y$ is the complex space $\psi^{-1}(0)$, with the naturally induced structure sheaf (which may contain nilpotent elements).
(ii) The two-jet of $\psi$ at the origin is given by the combination of cup product and bracket: $$ H^1(\operatorname{End} \mathscr{E}) \otimes H^1(\operatorname{End} \mathscr{E}) \to H^2(\operatorname{End}_0 \mathscr{E}). $$
(iii) If $H^0(\operatorname{End}_0 \mathscr{E})$ is zero, so that the groups $\operatorname{Aut} \mathscr{E}$ is equal to the scalars $\mathbb{C}^*$, then $Y$ is a universal deformation, and a neighbourhood of $[\mathscr{E}]$ in the quotient space $\mathscr{A}^{1,1}/\mathscr{G}^c$ (in the quotient topology) is homeomorphic to the space underlying $Y$. More generally, if $\operatorname{Aut} \mathscr{E}$ is a reductive group we can choose $\psi$ to be $\operatorname{Aut} \mathscr{E}$ equivariant, so $\operatorname{Aut} \mathscr{E}$ acts on $Y$ and a neighbourhood in the quotient is modelled on $Y/{\operatorname{Aut} \mathscr{E}}$ (which may not be Hausdorff).
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I have trouble understanding part (iii) where it is stated that $Y/\mathrm{Aut}(\mathscr{E})$ is the local model of the moduli space of holomorphic vector bundle if $\mathrm{Aut}(\mathscr{E})$ is reductive. Here, $Y$ is the Kuranishi space, i.e., the parameter space of the Kuranishi family.
If the automorphism group is compact, I can fill in the missing details. How do I pass from compactness to reductiveness? Can anybody point me to the right directions or some references. Thanks.
Notations:
$\mathscr{E}$: the holomorphic vector bundle.
$\mathscr{A}^{1,1}$: the space of connections whose curvatures are $(1,1)$-forms.
$\mathscr{G}^c$: the group of (smooth) automorphism of the underlying smooth vector bundle of $\mathscr{E}$.