A generalization of Hartog's extension of line bundles

TL;DR

This paper extends Hartog's theorem by proving that holomorphic line bundles over certain q-convex subsets of complex manifolds can be uniquely extended to the entire manifold, generalizing previous results for q-complete cases.

Contribution

It generalizes Hartog's extension theorem for line bundles to a broader class of q-convex manifolds with corners, under specific dimensional conditions.

Findings

Holomorphic line bundles extend uniquely over q-convex subsets.

Generalization of Hartog's extension to q-convex manifolds with corners.

Extension results hold for manifolds with dimension n ≥ 4 and 1 ≤ q ≤ n-3.

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 $n \geq 3q$.