Arkiv för Matematik

  • Ark. Mat.
  • Volume 54, Number 2 (2016), 299-319.

The Hartogs extension theorem for holomorphic vector bundles and sprays

Rafael B. Andrist, Nikolay Shcherbina, and Erlend F. Wold

Full-text: Open access


We give a detailed proof of Siu’s theorem on extendibility of holomorphic vector bundles of rank larger than one, and prove a corresponding extension theorem for holomorphic sprays. We apply this result to study ellipticity properties of complements of compact subsets in Stein manifolds. In particular we show that the complement of a closed ball in Cn,n3, is not subelliptic.


E. F. Wold is supported by grant NFR-209751/F20 from the Norwegian Research Council.

Article information

Ark. Mat., Volume 54, Number 2 (2016), 299-319.

Received: 4 December 2014
Revised: 1 July 2015
First available in Project Euclid: 30 January 2017

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

2015 © Institut Mittag-Leffler


Andrist, Rafael B.; Shcherbina, Nikolay; Wold, Erlend F. The Hartogs extension theorem for holomorphic vector bundles and sprays. Ark. Mat. 54 (2016), no. 2, 299--319. doi:10.1007/s11512-015-0226-y.

Export citation


  • Andersén, E., Volume-preserving automorphisms of Cn, Complex Var. Theory Appl. 14 (1990), 223–235.
  • Andersén, E. and Lempert, L., On the group of holomorphic automorphisms of Cn, Invent. Math. 110 (1992), 371–388.
  • Fornæss, J. E., Sibony, N. and Wold, E. F., Q-complete domains with corners in Pn and extension of line bundles, Math. Z. 273 (2013), 589–604.
  • Forstnerič, F. and Rosay, J.-P., Approximation of biholomorphic mappings by automorphisms of Cn, Invent. Math. 112 (1993), 323–349.
  • Forstnerič, F. and Rosay, J.-P., Erratum: “Approximation of biholomorphic mappings by automorphisms of Cn”, Invent. Math. 118 (1994), 573–574.
  • Forstnerič, F., The Oka principle for sections of subelliptic submersions, Math. Z. 241 (2002), 527–551.
  • Forstnerič, F., Stein manifolds and holomorphic mappings, in The Homotopy Principle in Complex Analysis, Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge. A Series of Modern Surveys in Mathematics [Results in Mathematics and Related Areas. 3rd Series. A Series of Modern Surveys in Mathematics] 56, p. xii+489, Springer, Heidelberg, 2011.
  • Forstnerič, F., Extending holomorphic mappings from subvarieties in Stein manifolds, Ann. Inst. Fourier (Grenoble) 55 (2005), 733–751. (English, with English and French summaries.)
  • Forstnerič, F., Runge approximation on convex sets implies the Oka property, Ann. of Math. (2) 163 (2006), 689–707.
  • Forstnerič, F., Oka manifolds, C. R. Math. Acad. Sci. Paris 347 (2009), 1017–1020. (English, with English and French summaries.)
  • Forstnerič, F. and Ritter, T., Oka properties of ball complements, Math. Z. 277 (2014), 325–338.
  • Gromov, M., Oka’s principle for holomorphic sections of elliptic bundles, J. Amer. Math. Soc. 2 (1989), 851–897.
  • Grauert, H., Holomorphe Funktionen mit Werten in komplexen Lieschen Gruppen, Math. Ann. 133 (1957), 450–472. (German).
  • Henkin, G. and Leiterer, J., The Oka–Grauert principle without induction over the base dimension, Math. Ann. 311 (1998), 71–93.
  • Ivashkovich, S., Bochner–Hartogs type extension theorem for roots and logarithms of holomorphic line bundles, Tr. Mat. Inst. Steklova 279 (2012), 269–287.
  • Siu, Y.-T., A Hartogs type extension theorem for coherent analytic sheaves, Ann. of Math. (2) 93 (1971), 166–188.
  • Siu, Y.-T., Techniques of extension of analytic objects, Lecture Notes in Pure and Applied Mathematics 8, p. iv+256, Dekker, New York, 1974.
  • Stout, E. L., Polynomial convexity, Progress in Mathematics 261, p. xii+439, Birkhäuser Boston, Boston, MA, 2007.