Skip to main content
SpringerLink
Log in

Cartan Theorems for Stein Manifolds Over a Discrete Valuation Base

  • Published:
The Journal of Geometric Analysis Aims and scope Submit manuscript

Abstract

Let X be a complex manifold, let A be a topological discrete valuation ring, and write for the sheaf of functions on X with values in A. We prove Cartan theorems A and B for coherent -modules, when X is a Stein manifold and A satisfies some requirements like being a nuclear direct limit of Banach algebras. The result is motivated by questions in the work of the second author with Kashiwara in the proof of the codimension-three conjecture for holonomic microdifferential systems.

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

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Bungart, L.: Holomorphic functions with values in locally convex spaces and applications to integral formulas. Trans. Am. Math. Soc. 111, 317–344 (1964)

    Article  MathSciNet  MATH  Google Scholar 

  2. Frisch, J., Guenot, J.: Prolongement de faisceaux analytiques cohérents. Invent. Math. 7, 321–343 (1969)

    Article  MathSciNet  MATH  Google Scholar 

  3. Grauert, H., Remmert, R.: Coherent Analytic Sheaves, Grundlehren der Mathematischen Wissenschaften 265, p. xviii+249. Springer, Berlin (1984)

    Google Scholar 

  4. Grothendieck, A.: Sur certains espaces de fonctions holomorphes. I. (French). J. Reine Angew. Math. 192, 35–64 (1953)

    MathSciNet  MATH  Google Scholar 

  5. Hörmander, L.: An Introduction to Complex Analysis in Several Variables, 3rd edn. North-Holland Mathematical Library 7, p. xii\(+\)254. North-Holland, Amsterdam (1990)

  6. Kashiwara, M., Vilonen, K.: Microdifferential systems and the codimension-three conjecture. Ann. Math. 180, 573–620 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  7. Köthe, G.: Topological Vector Spaces. II. Grundlehren der Mathematischen Wissenschaften, 237, p. xii+331. Springer, New York (1979)

    Google Scholar 

  8. Krantz, S.: Function Theory of Several Complex Variables, 2nd edn. AMS Chelsea Publishing, Providence (2001)

    MATH  Google Scholar 

  9. Pietsch, A.: Nuclear locally Convex Spaces, Ergebnisse der Mathematik und ihrer Grenzgebiete, vol. 66. Springer, New York (1972)

    Google Scholar 

  10. Schapira, P.: Microdifferential Systems in the Complex Domain, Grundlehren der Mathematischen Wissenschaften, vol. 269. Springer, Berlin (1985)

    Book  Google Scholar 

  11. Serre, J.P.: Géométrie algébrique et géométrie analytique. Ann. Inst. Fourier, Grenoble 6, 1–42 (1955–1956)

  12. Siu, Y.-T.: Extending coherent analytic sheaves. Ann. Math. 90, 108–143 (1969)

    Article  MathSciNet  MATH  Google Scholar 

  13. Trautmann, G.: Ein KontinuitŠtssatz für die Fortsetzung kohärenter analytischer Garben. Arch. Math. (Basel) 18, 188–196 (1967)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics and Statistics, University of Helsinki, P.O.Box 68, 00014, Helsinki, Finland

    Jari Taskinen & Kari Vilonen

  2. School of Mathematics and Statistics, University of Melbourne, Melbourne, VIC, 3010, Australia

    Kari Vilonen

Authors
  1. Jari Taskinen
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Kari Vilonen
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Jari Taskinen.

Additional information

Jari Taskinen was supported in part by the Academy of Finland and the Väisälä Foundation. Kari Vilonen was supported in part by NSF Grants DMS-1402928 & DMS-1069316, the Academy of Finland, the ARC Grant DP150103525, the Humboldt Foundation, and the Simons Foundation.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Taskinen, J., Vilonen, K. Cartan Theorems for Stein Manifolds Over a Discrete Valuation Base. J Geom Anal 29, 577–615 (2019). https://doi.org/10.1007/s12220-018-0012-8

Download citation

  • Received:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s12220-018-0012-8

Keywords

Mathematics Subject Classification

Search

Navigation