Vtyjkyuk

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 $H^1(operatornameEnd mathscrE)$ to $H^2(operatornameEnd_0 mathscrE)$, with $psi$ and its derivative both vanishing at $0$, and a versal deformation of $mathscrE$ 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:
$$
H^1(operatornameEnd mathscrE)
otimes H^1(operatornameEnd mathscrE)
to
H^2(operatornameEnd_0 mathscrE).
$$

(iii)
If $H^0(operatornameEnd_0 mathscrE)$ is zero, so that the groups $operatornameAut mathscrE$ is equal to the scalars $mathbbC^*$, then $Y$ is a universal deformation, and a neighbourhood of $[mathscrE]$ in the quotient space $mathscrA^1,1/mathscrG^c$ (in the quotient topology) is homeomorphic to the space underlying $Y$.
More generally, if $operatornameAut mathscrE$ is a reductive group we can choose $psi$ to be $operatornameAut mathscrE$ equivariant, so $operatornameAut mathscrE$ acts on $Y$ and a neighbourhood in the quotient is modelled on $Y/operatornameAut mathscrE$ (which may not be Hausdorff).

(Original scanned image here.)

I have trouble understanding part (iii) where it is stated that $Y/mathrmAut(mathscrE)$ is the local model of the moduli space of holomorphic vector bundle if $mathrmAut(mathscrE)$ 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:

$mathscrE$: the holomorphic vector bundle.

$mathscrA^1,1$: the space of connections whose curvatures are $(1,1)$-forms.

$mathscrG^c$: the group of (smooth) automorphism of the underlying smooth vector bundle of $mathscrE$.