On holomorphic vector bundles over linearly concave manifolds
Let X be an open subset of the complex projective space \({\mathbb{P}}^ n\) such that, for each \(z\in X\), there is a projective line L in \({\mathbb{P}}^ n\) with \(z\in L\subset X\), and let E be a holomorphic vector bundle of generic splitting type \(a_ 1\geq...\geq a_ r\) on X. The paper deals with the relation between the global (on X) splitting behaviour of E and the splitting behaviour of E in certain infinitesimal neighbourhoods of the generic lines of E. For example, the following theorem is proved: Suppose X is simply connected and, for all \(z\in X\), the set of lines L with \(z\in L\subseteq X\) is connected. Then \(E\cong {\mathcal O}(a_ 1)\oplus...\oplus {\mathcal O}(a_ r)\) over X if and only if this is the case over the \((a_ 1-a_ r+2)th\) infinitesimal neighbourhoods of the generic lines of E. For \(a_ 1=a_ r\) this is known, because then the curvature of the corresponding (via Radon-Penrose-Ward transform) Yang-Mills connection can be identified with the family of restrictions of E to the second-order infinitesimal neighbourhoods of the generic lines [cf. \textit{G. M. Khenkin}, Dokl. Akad. Nauk SSSR 255, 844-847 (1980) and \textit{Yu. I. Manin}, Itogi Nauki Tekh., Ser. Sovrem. Probl. Mat. 17, 3-55 (1981; Zbl 0486.53051)]. The proof of the general case uses a generalization of the Radon-Penrose-Ward transform, where instead of connections more general differential operators are obtained.
splitting behaviour in certain infinitesimal neighbourhoods
scientific article; zbMATH DE number 3913298