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

Sergey Feklistov

TL;DR

The paper addresses Hartogs extension for holomorphic fiber bundles φ: X → Y with noncompact, (1,0)-compactifiable fibers F. It combines the Leray spectral sequence for cohomology with compact supports and a density lemma in the QDFS topology to derive vanishing results and transfer Hartogs-type extendability from fibers to the total space, culminating in a cohomological criterion that ties Hartogs phenomena to finite-dimensional $H^{1}_{c}(X,\mathcal{O}_{X})$. By developing canonical topologies on cohomology groups and a topological Künneth framework, the work provides precise control over stalks and inductive limits, enabling sharp statements for Stein and non-Stein fibers. A central contribution is showing that if F is $(1,0)$-compactifiable with dim $F>1$ and F itself has the Hartogs phenomenon, then the total space X inherits the Hartogs property, with concrete instances in toric and semiabelian-fiber bundle settings. Overall, the results unify and extend classical Hartogs extension theory to holomorphic fiber bundles, offering new vanishing theorems, density tools, and a cohomological criterion that can be applied to a broad class of fibered complex spaces.

Abstract

We use the Leray spectral sequence for the sheaf cohomology groups with compact supports to obtain a vanishing result. The stalks of sheaves $R^{\bullet}φ_{!}\mathcal{O}$ 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.

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

TL;DR

The paper addresses Hartogs extension for holomorphic fiber bundles φ: X → Y with noncompact, (1,0)-compactifiable fibers F. It combines the Leray spectral sequence for cohomology with compact supports and a density lemma in the QDFS topology to derive vanishing results and transfer Hartogs-type extendability from fibers to the total space, culminating in a cohomological criterion that ties Hartogs phenomena to finite-dimensional . By developing canonical topologies on cohomology groups and a topological Künneth framework, the work provides precise control over stalks and inductive limits, enabling sharp statements for Stein and non-Stein fibers. A central contribution is showing that if F is -compactifiable with dim and F itself has the Hartogs phenomenon, then the total space X inherits the Hartogs property, with concrete instances in toric and semiabelian-fiber bundle settings. Overall, the results unify and extend classical Hartogs extension theory to holomorphic fiber bundles, offering new vanishing theorems, density tools, and a cohomological criterion that can be applied to a broad class of fibered complex spaces.

Abstract

We use the Leray spectral sequence for the sheaf cohomology groups with compact supports to obtain a vanishing result. The stalks of sheaves for the structure sheaf on the total space of a holomorphic fiber bundle 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 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.
Paper Structure (17 sections, 25 theorems, 79 equations)

This paper contains 17 sections, 25 theorems, 79 equations.

Table of Contents

  1. Introduction
  2. The Leray spectral sequence for sheaf cohomology groups with compact supports
  3. The spectral sequence of composed functors
  4. The Leray spectral sequenses
  5. Differential-geometric description of $R^{i}\phi_{!}\mathcal{O}_{X}$
  6. Canonical topologies on the sheaf cohomology groups and the topological Künnet formula
  7. FS and DFS type spaces and properties
  8. QFS-topology on $H^{\bullet}(X,\mathcal{F})$
  9. QDFS-topology on $H^{\bullet}_{c}(X,\mathcal{F})$
  10. QDFS-topology on $H^{\bullet}_{c}(Z_y,\mathcal{F}\mid_{Z_y})$
  11. Inductive topology on $H^{\bullet}_{c}(F_y,\mathcal{F}\mid_{F_y})$ with respect to the pair of spaces
  12. The duality theorems
  13. The nuclear spaces and topological Künnet formula
  14. On separatedness of the space $H^{\bullet}_{c}(F_{y},\mathcal{O}_{X}\mid_{F_{y}})$
  15. Density lemma for holomorphic fiber bundle and Stein fibers case
  16. ...and 2 more sections

Key Result

Theorem 1.1

Let $\phi\colon X\to Y$ be a holomorphic fibre bundle with $(1,0)$-com-pac-ti-fiable fiber $F$, $\dim F>1$. If $F$ admits the Hartogs phenomenon, then $X$ also admits the Hartogs phenomenon.

Theorems & Definitions (58)

  • Definition 1.1
  • Theorem 1.1
  • Theorem 2.1
  • Theorem 2.2
  • Definition 2.1
  • Remark 2.1
  • Definition 2.2
  • Remark 2.2
  • Lemma 2.1
  • proof
  • ...and 48 more