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A generalization of Hartog's extension of line bundles

Youssef Alaoui

TL;DR

The paper generalizes Hartogs' extension phenomenon for holomorphic line bundles to complex manifolds of dimension $n\ge 4$ that admit a $q$-convex with corners function with $1\le q\le n-3$, establishing that the restriction map $H^{p}(X, \mathcal{O}^{*}) \rightarrow H^{p}(Y, \mathcal{O}^{*})$ is bijective for $p=0,1,2$ and injective for $p=3$ where $Y=\{f>f(\xi_0)\}$. The proof combines local cohomology vanishing results for $q$-convex-with-corners domains, the Grothendieck spectral sequence, and Mittag-Leffler arguments to extend local extensions to a global setting without requiring $f$ to be exhaustive. This generalizes the earlier result for $q$-complete-with-corners manifolds (when $n\ge 3q$) and clarifies the extendability of line bundles in terms of low-degree cohomology of $\mathcal{O}^{*}$. The work bridges Hartogs-type extension phenomena with cohomological machinery, offering a practical criterion for extending line bundles across sublevel sets defined by $f$ and broadening the scope beyond exhaustiveness assumptions.

Abstract

In this article, we prove that if $X$ is a complex manifold of dimension $n\geq 4$ such that there exists a $q$-convex with corners function $f\in F_{q}(X)$, then every holomorphic line bundle over $\{f>c\}$ extends uniquely to $X$ if $1\leq q\leq n-3$. This generalizes a well-known result obtained in \cite{ref5} for $q$-complete with corners complex manifolds with a corresponding exhaustion function $f \in F_{q}(X)$, when $n \geq 3q$.

A generalization of Hartog's extension of line bundles

TL;DR

The paper generalizes Hartogs' extension phenomenon for holomorphic line bundles to complex manifolds of dimension that admit a -convex with corners function with , establishing that the restriction map is bijective for and injective for where . The proof combines local cohomology vanishing results for -convex-with-corners domains, the Grothendieck spectral sequence, and Mittag-Leffler arguments to extend local extensions to a global setting without requiring to be exhaustive. This generalizes the earlier result for -complete-with-corners manifolds (when ) and clarifies the extendability of line bundles in terms of low-degree cohomology of . The work bridges Hartogs-type extension phenomena with cohomological machinery, offering a practical criterion for extending line bundles across sublevel sets defined by and broadening the scope beyond exhaustiveness assumptions.

Abstract

In this article, we prove that if is a complex manifold of dimension such that there exists a -convex with corners function , then every holomorphic line bundle over extends uniquely to if . This generalizes a well-known result obtained in \cite{ref5} for -complete with corners complex manifolds with a corresponding exhaustion function , when .
Paper Structure (2 sections, 5 theorems, 20 equations)

This paper contains 2 sections, 5 theorems, 20 equations.

Table of Contents

  1. Introduction
  2. Proof of the theorem

Key Result

Theorem 1

Let $X$ be a complex manifold of dimension $n\geq 4$ and $\xi_{0}\in X$. Suppose that there exists a $q$-convex with corners function $f\in C^{0}(X )$ with $1\leq q\leq n-3$, and let $Y=\{z\in X: f(z)>f(\xi_{0})\}$. Then the restriction map is bijective if $p=0, 1, 2$ and injective if $p=3$.

Theorems & Definitions (9)

  • Theorem
  • Lemma 1
  • Lemma 2
  • proof
  • Lemma 3
  • proof
  • Lemma 4
  • proof
  • proof