Hermitian geometry and involutive algebras
Let \(M\) be a connected complex manifold, \({\mathcal H}\) a separable complex Hilbert space and \(Gr(n,{\mathcal H})\) the Grassmann manifold, that is, the set of all n-dimensional subspaces of \({\mathcal H}\). The aim of the paper is to link various properties of holomorphic maps from \(M\) into \(Gr(n,{\mathcal H})\) to the theory of hermitian holomorphic vector bundles over \(M\) of rank \(n\), with real-analytic metrics. The first main result (the congruence theorem) asserts in essence that two holomorphic maps from \(M\) into \(Gr(n,{\mathcal H})\) which agree in an osculation sense up to order n, have images which differ from each other only by a global unitary mapping of \({\mathcal H}\) into itself. Let \(H0_ n(M)\) be the category of all hermitian holomorphic vector bundles over M of rank n, with real analytic metrics. The second main result of the paper (the equivalence theorem) gives an answer to the equivalence problem in \(H0_ n(M)\). One proves that two objects in \(H0_ n(M)\) which are related in a pointwise sense up to order \(n\), are locally equivalent. Along the way one solves also the equivalence problem for hermitian smooth vector bundles with metric preserving linear connections. The first instance of both theorems was due to \textit{M. J. Cowen} and \textit{R. G. Douglas} [Acta Math. 141, 187-261 (1978; Zbl 0427.47016)] in the case when M is an open subset of the complex plane. Our approach is different and involves some results and techniques from the theory of involutive algebras.
scientific article; zbMATH DE number 3859790