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 $R^{\bullet}ϕ_{!}\mathcal{O}$ for the structure sheaf $\mathcal{O}$ on the total space of a holomorphic fiber bundle $ϕ$ has canonical topology structures. Using the standard \vCech 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 $R^{1}ϕ_{!}\mathcal{O}$ 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.