Holomorphic extension in holomorphic fiber bundles with (1, 0)-compactifiable fiber

We use the Leray spectral sequence for the sheaf cohomology groups with compact supports to obtain a vanishing result. The stalks of sheaves \(R^{\bullet }\phi _{!}\mathcal {O}\) for the structure sheaf \(\mathcal {O}\) on the total space of a holomorphic fiber bundle \(\phi \) has canonical topology structures. Using the standard Čech argument we prove a density lemma for QDFS-topology on this stalks. In particular, we obtain a vanishing result for holomorphic fiber bundles with Stein fibers. Using Künnet formulas, properties of an inductive topology (with respect to the pair of spaces) on the stalks of the sheaf \(R^{1}\phi _{!}\mathcal {O}\) and a cohomological criterion for the Hartogs phenomenon we obtain the main result on the Hartogs phenomenon for the total space of holomorphic fiber bundles with (1, 0)-compactifiable fibers.