# A splitting lemma for coherent sheaves

## TL;DR

This paper introduces a splitting lemma for coherent sheaves that simplifies the proof of Oka principles by extending existing techniques and providing a new lifting lemma for transition maps.

## Contribution

It presents a novel splitting lemma and a lifting lemma for transition maps, advancing the methods used in the proof of Oka principles and related complex geometry results.

## Key findings

- Simplifies proofs of Oka principles for admissible pairs.
- Extends Gromov's results on elliptic submersions.
- Provides a new technique for gluing local sections of analytic sheaves.

## Abstract

The presented splitting lemma extends the techniques of Gromov and Forstneri\v{c} to glue local sections of a given analytic sheaf, a key step in the proof of all Oka principles. The novelty on which the proof depends is a lifting lemma for transition maps of coherent sheaves, which yields a reduction of the proof to the work of Forstneri\v{c}. As applications we get shortcuts in the proofs of Forster and Ramspott's Oka principle for admissible pairs and of the interpolation property of sections of elliptic submersions, an extension of Gromov's results obtained by Forstneri\v{c} and Prezelj.