The Thullen type extension theorem for holomorphic vector bundles with \(L^ 2\)-bounds on curvature
type: Item
instance of: scholarly article
title: The Thullen type extension theorem for holomorphic vector bundles with \(L^ 2\)-bounds on curvature
zbMATH Open document ID: 0858.32024
DOI: 10.1007/BF01444234
published in: Mathematische Annalen
publication date: 1997-03-19T00:00:00Z
full work available at URL: https://eudml.org/doc/183000
review text: Let \(X\) be a complex manifold and let \(Q\) be a closed nowhere dense subset of \(X\). The phenomenon of extendability of analytic objects from \(X\backslash Q\) to \(X\) is a central topic in modern analytic geometry. In particular, for analytic functions \(f\) there is a variety of extension theorems, all assuming a regularity of the set \(Q\) (e.g. \(Q\) is an analytic set), a dimension condition (e.g. \(\text{codim} (Q)>1)\) and/or a growth control of \(|f|\) near all points of \(Q\). The paper under review is a part of the author's PhD thesis at the Ruhr-University in Bochum. The objects to be extended this time are holomorphic Hermitian vector bundles; control is imposed on the set \(Q\) and on the growth of the curvature of the canonical holomorphic connection. To be more specific, we reproduce the main result: Main Theorem. Let \(X\) be a complex manifold and \(Q\) a closed subset of \(X\) of the form \(Q = \widetilde Q \backslash G\) where either i) \(\widetilde Q\) is an analytic set in \(X\), \(G\) is an open subset of \(X\) having nonempty intersection with every irreducible component of \(Q\) of complex codimension 1; or ii) \(\widetilde Q\) is a connected Lipschitz submanifold of real codimension 2 and \(G\) is an open subset of \(X\) having nonempty intersection with \(\widetilde Q\). Let \(E\) be a holomorphic Hermitian vector bundle over \(X\backslash Q\) with the curvature \(F\). If \(F\in L^2_{\text{loc}}(X)\), then \(E\) extends through \(Q\) as a reflexive sheaf \(\widetilde E\). If, in addition, \(F\in L^p_{\text{loc}} (X)\) with some \(p\geq 3\), then \(\widetilde E\) is a bundle outside \(Q_0\), which is an analytic set in \(X\) with \(\text{codim}_\mathbb{C} Q_0 > p\). The proof is rather technical. Roughly, the main idea is to prove Malgrange's nonlinear Dolbeault-Grothendieck Lemma with bounds. More specifically, these bounds are derived from Hörmander's celebrated \(L^p\)-estimates for the Cauchy-Riemann complex.
reviewed by: Mihai Putinar
Mathematics Subject Classification ID: 32Q20
Mathematics Subject Classification ID: 32D15
Mathematics Subject Classification ID: 32-02
zbMATH DE Number: 931970
zbMATH Keywords: curvature
zbMATH Keywords: extendability of analytic objects
zbMATH Keywords: holomorphic Hermitian vector bundles
author: Vsevolod Shevchishin
MaRDI profile type: Publication
cites work: Removable singularities for holomorphic vector bundles
cites work: Q5565773
cites work: Q5187597
cites work: Q4195061
cites work: Removable singularities of solutions of linear partial differential equations
cites work: Q3834309
cites work: The currents defined by analytic varieties
cites work: Q5565587
cites work: Q4011138
cites work: The genus-minimizing property of algebraic curves
cites work: Q5575760
cites work: Espaces Homogènes De Stein Des Groupes De Lie Complexes
cites work: Q5518064
cites work: Vector bundles on complex projective spaces
cites work: Q3263052
cites work: Ein neuer Beweis des Hartogsschen Hauptsatzes und seine Ausdehnung auf meromorphe Funktionen
cites work: The Oka-Grauert principle for the extension of holomorphic line bundles with integrable curvature
cites work: Q4876614
cites work: Extension of positive line bundles and meromorphic maps
cites work: Singular Sobolev connections with holonomy
cites work: Classification of singular Sobolev connections by their holonomy
cites work: Solutions to Yang—Mills equations that are not self-dual
cites work: Q4046376
cites work: A Thullen type extension theorem for positive holomorphic vector bundles
cites work: Extending coherent analytic sheaves
cites work: Ein Kontinuitätssatz für die Fortsetzung kohärenter analytischer Garben
cites work: Connections with \(L^ p \)bounds on curvature
cites work: Removable singularities in Yang-Mills fields
description: scientific article; zbMATH DE number 931970