Vanishing of quaternionic cohomology groups and applications

TL;DR

This paper investigates quaternionic cohomology groups related to slice-regular functions on axially symmetric domains, proving their vanishing and applying these results to solve Cousin problems, jet interpolation, and divisor principalization.

Contribution

It establishes the vanishing of quaternionic cohomology groups for axially symmetric domains and applies this to solve classical problems in quaternionic analysis.

Findings

Vanishing of cohomology groups with respect to axially symmetric coverings

Solutions to additive and multiplicative Cousin problems for slice-regular functions

Every divisor is principal in the studied setting

Abstract

We present solutions to additive and multiplicative Cousin problems formulated on an axially symmetric domain $\Omega \subset \mathbb H$ for slice--regular functions starting from the solutions for subclasses, namely slice--regular slice--preserving functions and functions in a given vectorial class. Consequently, we prove the vanishing of the corresponding cohomology groups with respect to axially symmetric open coverings (Theorems 1.1, 4.1, 4.2). The primary tool used in the proofs of these theorems is the existence of quaternionic Cartan coverings and Cartan's splitting lemmas. As an application, we prove a jet interpolation theorem (Theorem 1.2) and show that every divisor is principal (Theorem 5.1).