Holomorphic vector bundles on quadric hypersurfaces of infinite-dimensional projective spaces

Abstract

Here we prove the following result and a few related statements. Let $V$ be a Banach space with countable unconditional basis and the localizing property, $Q\subset P(V)$ a quadric hypersurface with finite-dimensional singular locus and $E$ a holomorphic vector bundle of finite rank on $Q$. Then $E\cong\oplus_{1\leq i\leq r}0_{Q}(a_{i})$ for some integers $a_{i}$ and $h^{1}(Q, E(t))=0$ for every integer $t$.