Cartan Theorems for Stein Manifolds Over a Discrete Valuation Base

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, access via your institution.

References

  1. 1.

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

    MathSciNet  Article  MATH  Google Scholar 

  2. 2.

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

    MathSciNet  Article  MATH  Google Scholar 

  3. 3.

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

    Google Scholar 

  4. 4.

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

    MathSciNet  MATH  Google Scholar 

  5. 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. 6.

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

    MathSciNet  Article  MATH  Google Scholar 

  7. 7.

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

    Google Scholar 

  8. 8.

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

    Google Scholar 

  9. 9.

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

    Google Scholar 

  10. 10.

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

    Google Scholar 

  11. 11.

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

  12. 12.

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

    MathSciNet  Article  MATH  Google Scholar 

  13. 13.

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

    MathSciNet  Article  MATH  Google Scholar 

Download references

Author information

Affiliations

Authors

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

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

Keywords

  • Cartan theorem A and B
  • Stein manifold
  • Coherent module
  • Codimension-three conjecture
  • Discrete valuation ring
  • Banach algebra
  • Inductive limit

Mathematics Subject Classification

  • Primary 32C35
  • Secondary 32W05
  • 46A13
  • 46H99