Flat bundles over some compact complex manifolds

TL;DR

This paper constructs specific examples of flat fiber bundles over compact complex manifolds, demonstrating complex geometric properties like lack of pseudoconvex neighborhoods and absence of nonconstant holomorphic functions.

Contribution

It provides explicit constructions of flat bundles with unusual complex geometric properties, expanding understanding of their behavior over various compact complex manifolds.

Findings

Total spaces lack pseudoconvex neighborhood basis

Existence of hyperconvex total spaces with no nonconstant holomorphic functions

Construction of Stein domains with non-pseudoconvex closures

Abstract

We construct examples of flat fiber bundles over the Hopf surface such that the total spaces have no pseudoconvex neighborhood basis, admit a complete K\"ahler metric, or are hyperconvex but have no nonconstant holomorphic functions. For any compact Riemannian surface of positive genus, we construct a flat $\mathbb P^1$ bundle over it and a Stein domain with real analytic bundary in it whose closure does not have pseudoconvex neighborhood basis. For a compact complex manifold with positive first Betti number, we construct a flat disc bundle over it such that the total space is hyperconvex but admits no nonconstant holomorphic functions.