Isotropies of Partial Connections and a Theorem of Morimoto

Abstract

Let E be a smooth complex vector bundle over a compact complex manifold M. The complex gauge group G of E is a non-normable complex Fréchet Lie group which acts smoothly on the affine Fréchet space C″ of all partial (0, 1)-connections in E . 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 G . 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 ∂ -operator of the bundle.