Skip to main content
SpringerLink
Log in

On the Density and the Volume Density Property

  • Conference paper
  • First Online:
Complex Analysis and Geometry

Part of the book series: Springer Proceedings in Mathematics & Statistics ((PROMS,volume 144))

Abstract

This article gives a short introduction into the notions of density property (DP) and volume density property (VDP). Moreover we develop an effective criterion of verifying whether a given X has VDP. As an application of this method we give a new proof of the basic fact that the product of two Stein manifolds with VDP admits VDP.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 84.99
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 109.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD 109.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Notes

  1. 1.

    Recall that an open subset U of X is Runge if any holomorphic function on U can be approximated by global holomorphic functions on X in the compact-open topology. Actually, for X Stein by Cartan’s Theorems A and B this definition implies more: for any coherent sheaf on X its section over U can be approximated in the compact-open topology by global sections.

  2. 2.

    In fact in the coming paper of the authors these results are extended to affine homogeneous spaces of linear algebraic groups. More, precisely any such a space different from \(\mathbb {C}\) or \((\mathbb {C}^*)^k\) has DP. Similarly, any such a space (including \(\mathbb {C}\) or \((\mathbb {C}^*)^k\)) equipped with a left invariant volume form has VDP.

References

  1. Andersén, E.: Complete vector fields on \(({\mathbb{C}}^*)^n\). Proc. Am. Math. Soc. 128(4), 1079–1085 (2000)

    Article  MATH  Google Scholar 

  2. Andersén, E.: Volume-preserving automorphisms of \({\mathbb{C}}^n\). Complex Var. Theory Appl. 14(1–4), 223–235 (1990)

    Article  MATH  Google Scholar 

  3. Andersén, E., Lempert, L.: On the group of holomorphic automorphisms of \({ C}^n\). Invent. Math. 110(2), 371–388 (1992)

    Article  MathSciNet  MATH  Google Scholar 

  4. Rafael, B.: Andrist, Stein spaces characterized by their endomorphisms. Trans. Am. Math. Soc. 363(5), 2341–2355 (2011)

    Article  MATH  Google Scholar 

  5. Andrist, R., Forstnerič, F., Ritter, T., Wold, E.F.: Proper holomorphic embeddings into Stein manifolds with the density property. arXiv: To appear in J. d’Analyse Math

    Google Scholar 

  6. Andrist, R.B., Wold, E.F.: Riemann surfaces in Stein manifolds with density property. arXiv:1106.4416

  7. Borell, S., Kutzschebauch, F.: Non-equivalent embeddings into complex Euclidean spaces. Int. J. Math. 17(9), 1033–1046 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  8. Buzzard, G.T., Fornæss, J.-E.: An embedding of \({\mathbb{C}}\) with hyperbolic complement. Math. Ann. 306(3), 539–546 (1996)

    Google Scholar 

  9. Derksen, H.: Frank Kutzschebauch Nonlinearizable holomorphic group actions. Math. Ann. 311(1), 41–53 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  10. Derksen, H., Kutzschebauch, F.: Global holomorphic linearization of actions of compact Lie groups on \({\mathbb{C} }^n\). Complex geometric analysis in Pohang (1997). Contemp. Math. 222, 201–210 (Amer. Math. Soc., Providence, RI, 1999)

    Google Scholar 

  11. Donzelli, F.: Algebraic density property of Danilov-Gizatullin surfaces. Math. Z. 272(3–4), 1187–1194 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  12. Donzelli, F., Dvorsky, A., Kaliman, S.: Algebraic density property of homogeneous spaces. Transform. Groups 15(3), 551–576 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  13. Forstnerič, F.: Stein Manifolds and Holomorphic Mappings, vol. 56 of Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. F. Springer, Heidelberg (2011)

    Google Scholar 

  14. Forstnerič, F., Rosay, J.-P.: Approximation of biholomorphic mappings by automorphisms of \({ C}^n\). Invent. Math. 112(2), 323–349 (1993)

    Article  MathSciNet  MATH  Google Scholar 

  15. Forstnerič, F., Ivarsson, B., Kutzschebauch, F., Prezelj, J.: An interpolation theorem for proper holomorphic embeddings. Math. Ann. 338(3), 545–554 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  16. Forstnerič, F., Wold, E.F.: Embeddings of infinitely connected planar domains into \({\mathbb{C}}^2\). Anal. PDE 6(2), 499–514 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  17. Forstnerič, F., Wold, E.F.: Bordered Riemann surfaces in \({\mathbb{C}}^2\). J. Math. Pures Appl. (9), 91 (2009) ((1), 100–114)

    Google Scholar 

  18. Forstnerič, F., Globevnik, J., Rosay, J.-P.: Nonstraightenable complex lines in \({\mathbb{C}}^2\). Ark. Math. 34(1), 97–101 (1996)

    Article  MATH  Google Scholar 

  19. Grauert, H., Remmert, R.: Theory of Stein Spaces. Translated from the German by Alan Huckleberry. Reprint of the: translation. Springer, Berlin, Classics in Mathematics (1979)

    Google Scholar 

  20. Peters, H., Wold, E.F.: Non-autonomous basins of attraction and their boundaries. J. Geom. Anal. 15(1), 123–136 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  21. Kaliman, S., Kutzschebauch, F.: On algebraic volume density property. Transform. Groups. arxiv:1201.4769 (to appear)

  22. Kaliman, S.: Frank Kutzschebauch Criteria for the density property of complex manifolds. Invent. Math. 172(1), 71–87 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  23. Kaliman, S., Kutzschebauch, F.: Density property for hypersurfaces \(uv=p({\bar{x}})\). Math. Z. 258(1), 115–131 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  24. Kaliman, S., Kutzschebauch, F.: On the present state of the Andersen-Lempert theory In: Affine Algebraic Geometry: The Russell Festschrift, pp. 85–122. Centre de Recherches Mathématiques. CRM Proceedings and Lecture Notes, vol. 54 (2011)

    Google Scholar 

  25. Kaliman, S., Kutzschebauch, F.: Algebraic volume density property of affine algebraic manifolds. Invent. Math. 181(3), 605–647 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  26. Kutzschebauch, F., Ramos Peon, A.: An Oka Principle for a Parametric Infinite Transitivity Property. arXiv:1401.0093

  27. Kutzschebauch, F., Wold, E.F.: Carleman approximation by holomorphic automorphisms of \({\mathbb{C}}^n\). J. Reine Angew, Math. (to appear) arXiv:1401.2842

  28. Leuenberger, M.: Complete holomorphic vector fields on affine surfaces. Ph.D. thesis, University of Bern (2015)

    Google Scholar 

  29. Kutzschebauch, F., Leuenberger, M., Liendo, A.: The algebraic density property for affine toric varieties. J. Pure Appl. Algebra 219(8), 3685–3700 (2015). arXiv:1402.2227

  30. Kutzschebauch, F., Lodin, S.: Holomorphic families of nonequivalent embeddings and of holomorphic group actions on affine space. Duke Math. J. 162(1), 49–94 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  31. Ritter, T.: A strong Oka principle for embeddings of some planar domains into \(\mathbb{C}\times \mathbb{C}^\ast \). J. Geom. Anal. 23(2), 571–597 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  32. Rosay, J.-P.: Automorphisms of \({\mathbb{C}}^n\), a survey of Andersén-Lempert theory and applications. Complex geometric analysis in Pohang. Contemp. Math., 222, 131–145 (1997) (Amer. Math. Soc., Providence, RI, 1999)

    Google Scholar 

  33. Varolin, D.: The density property for complex manifolds and geometric structures. II. Int. J. Math. 11(6), 837–847 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  34. Varolin, D.: The density property for complex manifolds and geometric structures. J. Geom. Anal. 11(1), 135–160 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  35. Varolin, D.: A general notion of shears, and applications. Michigan Math. J. 46(3), 533–553 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  36. Varolin, D.: Arpad Toth holomorphic diffeomorphisms of complex semisimple Lie groups. Invent. Math. 139(2), 351–369 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  37. Varolin, D., Toth, A.: Holomorphic diffeomorphisms of semisimple homogeneous spaces. Compos. Math. 142(5), 1308–1326 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  38. Wold, E.F.: Embedding subsets of tori properly into \({\mathbb{C}}^2\). Ann. Inst. Fourier (Grenoble) 57(5), 1537–1555 (2007)

    Google Scholar 

  39. Wold, E.F.: Embedding Riemann surfaces properly into \({\mathbb{C}}^2\). Int. J. Math. 17(8), 963–974 (2006)

    Google Scholar 

  40. Wold, E.F.: Proper holomorphic embeddings of finitely and some infinitely connected subsets of \({\mathbb{C}}\). Math. Z. 252(1), 1–9 (2006)

    Google Scholar 

  41. Wold, E.F.: A Fatou-Bieberbach domain in \({\mathbb{C}}^2\) which is not Runge. Math. Ann. 340(4), 775–780 (2008)

    Google Scholar 

  42. Wold, E.F.: A long \({\mathbb{C}}^2\) which is not Stein. Ark. Mat. 48(1), 207–210 (2010)

    Google Scholar 

Download references

Acknowledgments

This research was started during a visit of the first author to the University of Bern and continued during a visit of the second author to the University of Miami, Coral Gables. We thank these institutions for their generous support and excellent working conditions. The research of the first author was also partially supported by NSA Grant no. H982301010185 and the second author was also partially supported by Schweizerische Nationalfonds grants No. 200020-134876/1 and 200021-140235/1

Author information

Authors and Affiliations

  1. Department of Mathematics, University of Miami, Coral Gables, FL, 33124, USA

    Shulim Kaliman

  2. Mathematisches Institut, Universität Bern, Sidlerstr. 5, 3012, Bern, Switzerland

    Frank Kutzschebauch

Authors
  1. Shulim Kaliman
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Frank Kutzschebauch
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Frank Kutzschebauch .

Editor information

Editors and Affiliations

  1. Department of Mathematics, Università di Roma Tor Vergata, Rome, Italy

    Filippo Bracci

  2. Dept. of Mathematics Education, Kyungnam University, Changwon, Korea (Republic of)

    Jisoo Byun

  3. Université Joseph Fourier Grenoble, Institut Fourier, Grenoble, France

    Hervé Gaussier

  4. Graduate School of Mathematical Sciences, The University of Tokyo, Meguro-ku, Tokyo, Japan

    Kengo Hirachi

  5. Department of Mathematics, POSTECH, Pohang, Korea (Republic of)

    Kang-Tae Kim

  6. Department of Mathematics, Bergische Universität Wuppertal, Wuppertal, Nordrhein-Westfalen, Germany

    Nikolay Shcherbina

Rights and permissions

Reprints and permissions

Copyright information

© 2015 Springer Japan

About this paper

Cite this paper

Kaliman, S., Kutzschebauch, F. (2015). On the Density and the Volume Density Property. In: Bracci, F., Byun, J., Gaussier, H., Hirachi, K., Kim, KT., Shcherbina, N. (eds) Complex Analysis and Geometry. Springer Proceedings in Mathematics & Statistics, vol 144. Springer, Tokyo. https://doi.org/10.1007/978-4-431-55744-9_12

Download citation

Publish with us

Policies and ethics

Search

Navigation