E. Ballico

Projective Bundles on Infinite-Dimensional Complex Spaces

Abstract:
Let $V$ be a complex localizing Banach space with countable unconditional basis and $E$ a rank $r$ holomorphic vector bundle on ${\bf {P}}(V)$. Here we study the holomorphic embeddings of ${\bf {P}}(E)$ into products of projective spaces and the holomorphic line bundles on ${\bf {P}}(E)$. In particular we prove that if $r \ge 3$, then $H^1({\bf {P}}(E),L) = 0$ for every holomorphic line bundle $L$ on ${\bf {P}}(E)$.

Keywords:
Infinite-dimensional projective space, complex Banach manifold, holomorphic vector bundle, holomorphic line bundle, localizing Banach space, Banach space with countable unconditional basis.

MSC 2000: 32K05, 14N05.