Isotropies of Partial Connections and a Theorem of Morimoto

Let  be a smooth complex vector bundle over a compact complex manifold M. The complex gauge group  of  is a non-normable complex Fréchet Lie group which acts smoothly on the affine Fréchet space C" of all partial (0, 1)-connections in . The paper establishes a structural result for the isotropies of this action. Specifically, via methods involving the C^∞-topology of the gauge group, the isotropies are shown to be finite dimensional closed embedded complex Lie subgroups of . As an application, a new proof of an earlier result of Morimoto on the finite dimensionality of the group of holomorphic bundle automorphisms of a holomorphic vector bundle is given. The new proof uncovers additional information, namely, that this group is naturally realized as an extension of a closed Lie subgroup of the holomorphic transformations of the base M by the isotropy of the given <e15>∂</e15>-operator of the bundle.Copyright 1993, 1999 Academic Press