Isotropies of partial connections and a theorem of Morimoto (Q2367834)

Let \(E\) be a smooth complex vector bundle over a compact complex manifold \(M\). The complex gauge group \(G\) of \(E\) is a complex Fréchet Lie group which acts smoothly on the affine Fréchet space \(C''\) of all partial \((0,1)\)-connections in \(E\). The author shows, using the \(C^ \infty\)- topology of \(G\), that the isotropy subgroups for this action are finite- dimensional closed embedded Lie subgroups. As an application, he gives a new proof of a theorem of \textit{A. Morimoto} [Nagoya Math. J. 13, 157-168 (1958; Zbl 0107.286)] which states that the group of automorphisms of a holomorphic vector bundle on a compact complex manifold is a finite- dimensional complex Lie group. \{It should be noted that, contrary to present-day convention, the automorphisms considered by Morimoto are allowed to act non-trivially on \(M\); moreover Morimoto stated and proved his theorem for automorphisms of any holomorphic principal fibre bundle. Note also that the author calculates isotropies for all points of \(C''\), not just those which determine holomorphic structures on \(E\)\}.