The Levi problem over generalized Hirzebruch manifolds

TL;DR

This paper reviews classical techniques for the Levi problem with symmetries and applies them to new cases involving generalized Hirzebruch manifolds and certain Hopf surfaces, expanding the known solutions.

Contribution

It extends classical methods to solve the Levi problem in new geometric contexts, specifically generalized Hirzebruch manifolds and non-diagonal Hopf surfaces.

Findings

Successfully solved the Levi problem for generalized Hirzebruch manifolds.

Extended classical methods to new complex geometric settings.

Provided explicit solutions in previously unaddressed cases.

Abstract

We review classical methods to solve the Levi problem in the presence of symmetries, established by Hirschowitz and by Grauert-Remmert-Ueda. We then illustrate these methods by solving the Levi problem in some new situations, namely generalized Hirzebruch manifolds and primary Hopf surfaces of non-diagonal type.