**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.