On the cohomology groups of holomorphic Banach bundles

TL;DR

This paper investigates the relationship between cohomology groups of holomorphic Banach bundles over compact complex manifolds, showing that finite dimensionality is preserved under certain perturbations.

Contribution

It introduces the concept of compact perturbations of holomorphic Banach bundles and proves the invariance of finite dimensionality of cohomology groups under such perturbations.

Findings

Finite dimensionality of cohomology groups is preserved under compact perturbations.

Provides a more precise relationship between the cohomology groups of perturbed bundles.

Establishes a new framework for understanding cohomology stability in infinite-dimensional bundle settings.

Abstract

We consider a compact complex manifold $M$, and introduce the notion of two holomorphic Banach bundles $E,F$ over $M$ being compact perturbations of one another. Given two such bundles we show that if the cohomology groups $H^q(M,E)$ are finite dimensional then so are the cohomology groups $H^q(M,F)$; as well as a more precise result in the same spirit.