  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.
