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

TL;DR

This paper investigates the extension of holomorphic functions in fiber bundles with special fibers, establishing conditions under which the Hartogs phenomenon occurs using spectral sequences and sheaf cohomology.

Contribution

It introduces new vanishing results and cohomological criteria for the Hartogs phenomenon in holomorphic fiber bundles with (1,0)-compactifiable fibers.

Findings

Vanishing of certain sheaf cohomology groups for these bundles

Density lemma for QDFS-topology on sheaf stalks

Main result on Hartogs phenomenon for bundles with (1,0)-compactifiable fibers

Abstract

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 \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 K\"unnet 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.