Weakly 1-completeness of holomorphic fiber bundles over compact Kähler manifolds

Diederich and Ohsawa (Publ. Res. Inst. Math. Sci. 21 (1985), no. 4, 819–833) proved that every disc bundle over a compact Kähler manifold is weakly 1-complete. In this paper, under certain conditions, we generalize this result to the case of fiber bundles over compact Kähler manifolds whose fibers are bounded symmetric domains. In particular, if the representation related to the fiber bundle is reductive, then it has a plurisubharmonic exhaustion function. If the bundle is obtained by the diagonal action on the product of bounded symmetric domains, we are able to show that it is hyperconvex.