Union of linear spaces in infinite-dimensional projective spaces and holomorphic vector bundles
Let \(V\) be a complex Banach space (of infinite dimension) and \(X\subset \mathbb P(V)\) the transversal union of linear subspaces which all have the same finite codimension. More precisely: \(V\) is supposed to be a localizing Banach space with an unconditional countable basis (For these technicalities, see Lempert's papers). The author shows that any holomorphic vector bundle \(E\) of finite rank on \(X\) satisfies \(H^1(X,E)=0\) and is the direct sum of line bundles of the form \({\mathcal O}_X(n)\). The proof is based on the analogous result for \(X=\mathbb P(V)\) which was shown by \textit{L. Lempert} [J. Am. Math. Soc. 11, 485-520 (1998; Zbl 0904.32014)].
scientific article; zbMATH DE number 1997683