A splitting theorem for holomorphic Banach bundles

TL;DR

This paper proves a splitting theorem for holomorphic Banach bundles over compact complex manifolds, showing such bundles decompose into finite rank and trivial parts under certain cohomological conditions.

Contribution

It extends Grothendieck's splitting theorem to a class of Banach bundles that are compact perturbations of trivial bundles, under the condition that H^1(X, O)=0.

Findings

Holomorphic Banach bundles split into finite rank and trivial parts.

The splitting holds when H^1(X, O)=0.

The result generalizes classical splitting theorems to infinite-dimensional bundles.

Abstract

This paper is motivated by Grothendieck's splitting theorem. In the 1960s, Gohberg generalized this to a class of Banach bundles. We consider a compact complex manifold $X$ and a holomorphic Banach bundle $E \to X$ that is a compact perturbation of a trivial bundle in a sense recently introduced by Lempert. We prove that $E$ splits into the sum of a finite rank bundle and a trivial bundle, provided $H^{1}(X, \O)=0$.