Weakly 1-completeness of holomorphic fiber bundles over compact K\"ahler manifolds

TL;DR

This paper generalizes a 1985 result by showing that certain fiber bundles over compact Kähler manifolds are weakly 1-complete or hyperconvex, extending the class of known complex geometric structures.

Contribution

It extends the weak 1-completeness result to fiber bundles with bounded symmetric domain fibers and demonstrates hyperconvexity for bundles formed by diagonal actions.

Findings

Fiber bundles with bounded symmetric domain fibers are weakly 1-complete under certain conditions.

Bundles formed by diagonal actions on products of irreducible bounded symmetric domains are hyperconvex.

Generalization of Diederich and Ohsawa's 1985 result to broader classes of fiber bundles.

Abstract

In 1985 Diederich and Ohsawa proved that every disc bundle over a compact K\"ahler manifold is weakly 1-complete. In this paper, under certain conditions we generalize this result to the case of fiber bundles over compact K\"ahler manifolds whose fibers are bounded symmetric domains. Moreover if the bundle is obtained by the diagonal action on the product of irreducible bounded symmetric domains, we show that it is hyperconvex.