Some non-pseudoconvex domains with explicitly computable non-Hausdorff Dolbeault cohomology

TL;DR

This paper explicitly computes the Dolbeault cohomology of specific non-pseudoconvex domains in complex space, revealing their non-Hausdorff topological structure and identifying their reduced and indiscrete parts.

Contribution

It provides explicit calculations of Dolbeault cohomology for generalized Hartogs domains, highlighting their non-Hausdorff nature and decomposing the cohomology into reduced and indiscrete components.

Findings

Cohomology groups are non-Hausdorff topological vector spaces.

Explicit identification of reduced and indiscrete parts of cohomology.

Generalization of classical Hartogs figure domains.

Abstract

We explicitly compute the Dolbeault cohomologies of certain domains in complex space generalizing the classical Hartogs figure. The cohomology groups are non-Hausdorff topological vector spaces, and it is possible to identify the reduced (Hausdorff) and the indiscrete part of the cohomology.