A Counterexample to Hartogs’ Type Extension of Holomorphic Line Bundles



Consider a domain \(\varOmega \) in \(\mathbb {C}^n\) with \(n\geqslant 2\) and a compact subset \(K\subset \varOmega \) such that \(\varOmega \backslash K\) is connected. We address the problem whether a holomorphic line bundle defined on \(\varOmega \backslash K\) extends to \(\varOmega \). In 2013, Fornæss, Sibony, and Wold gave a positive answer in dimension \(n\geqslant 3\), when \(\varOmega \) is pseudoconvex and K is a sublevel set of a strongly plurisubharmonic exhaustion function. However, for K of general shape, we construct counterexamples in any dimension \(n\geqslant 2\). The key is a certain gluing lemma by means of which we extend any two holomorphic line bundles which are isomorphic on the intersection of their base spaces.


Hartogs’ extension Holomorphic line bundles Gluing lemma 

Mathematics Subject Classification

32A10 32L10 



The author addresses sincere thanks to Joël Merker for driving him to this problem and for useful discussions. The author also thanks an anonymous referee for pointing out minor mistakes in the first version.


  1. 1.
    Andreotti, A., Grauert, H.: Théorème de finitude pour la cohomologie des espaces complexes. Bull. Soc. Math. Fr. 90, 193–259 (1962)CrossRefMATHGoogle Scholar
  2. 2.
    Cartan, H.: Variétés analytiques complexes et cohomologie. In: Colloque sur les fonctions de plusieurs variables, tenu à Bruxelles, 1953, pp. 41–55. Georges Thone, Liège; Masson and Cie, Paris (1953)Google Scholar
  3. 3.
    Fornæss, J.E., Sibony, N., Wold, E.F.: \(Q\)-complete domains with corners in \(\mathbb{P}^n\) and extension of line bundles. Math. Z. 273(1–2), 589–604 (2013)MathSciNetCrossRefMATHGoogle Scholar
  4. 4.
    Ivashkovich, S.: Bochner–Hartogs type extension theorem for roots and logarithms of holomorphic line bundles. Tr. Mat. Inst. Steklova 279(Analiticheskie i Geometricheskie Voprosy Kompleksnogo Analiza), 269–287 (2012)Google Scholar
  5. 5.
    Merker, J., Porten, E.: A Morse-theoretical proof of the Hartogs extension theorem. J. Geom. Anal. 17(3), 513–546 (2007)MathSciNetCrossRefMATHGoogle Scholar
  6. 6.
    Michael Range, R.: Holomorphic Functions and Integral Representations in Several Complex Variables. Graduate Texts in Mathematics, vol. 108. Springer, New York (1986)CrossRefGoogle Scholar
  7. 7.
    Scheja, G.: Riemannsche Hebbarkeitssätze für Cohomologieklassen. Math. Ann. 144, 345–360 (1961)MathSciNetCrossRefMATHGoogle Scholar
  8. 8.
    Siu, Y.T.: Techniques of Extension of Analytic Objects. Lecture Notes in Pure and Applied Mathematics, vol. 8. Marcel Dekker, Inc., New York (1974)Google Scholar

Copyright information

© Mathematica Josephina, Inc. 2017

Authors and Affiliations

  1. 1.Département de Mathématiques D’orsay, Faculté des SciencesUniversité Paris-SudOrsay CedexFrance

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