Skip to main content
SpringerLink
Log in

On the nonvanishing of abstract Cauchy–Riemann cohomology groups

  • Published:
Mathematische Annalen Aims and scope Submit manuscript

Abstract

In this paper we prove infinite dimensionality of some local and global cohomology groups on abstract Cauchy–Riemann manifolds.

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. Andreotti, A., Fredricks, G., Nacinovich, M.: On the absence of Poincaré lemma in tangential Cauchy–Riemann complexes. Ann. Sc. Norm. Sup. Pisa 8, 365–404 (1981)

    MathSciNet  MATH  Google Scholar 

  2. Andreotti, A., Hill, C.D.: E.E. Levi convexity and the Hans Lewy problem. Part I. Ann. Sc. Norm. Super. Pisa 26, 325–363 (1972)

    MathSciNet  MATH  Google Scholar 

  3. Andreotti, A., Hill, C.D.: E.E. Levi convexity and the Hans Lewy problem. Part II. Ann. Sc. Norm. Super. Pisa 28, 747–806 (1972)

    MathSciNet  MATH  Google Scholar 

  4. Brinkschulte, J.: Local exactness of top-degree forms in abstract Cauchy–Riemann complexes. Manuscripta Math. 110, 137–143 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  5. Brinkschulte, J., Hill, C.D.: Malgrange’s vanishing theorem for weakly pseudoconcave \(CR\) manifolds. Manuscripta Math. 131, 503–506 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  6. Brinkschulte, J., Hill, C.D.: Obstructions to finite dimensional cohomology of abstract Cauchy-Riemann complexes. Ann. Sc. Norm. Super. Pisa (2015). doi:10.2422/2036-2145.201401_008

  7. Brinkschulte, J., Hill, C.D., Nacinovich, M.: The Poincaré lemma and local embeddability. B. U. M. I. 8, 393–398 (2003)

    MathSciNet  MATH  Google Scholar 

  8. Boutet de Monvel, L.: Intégration des équations de Cauchy–Riemann induites formelles. Sem. Goulaouic–Lions–Schwartz (1974–197)

  9. Hill, C.D., Nacinovich, M.: Embeddable CR manifolds with nonembeddable smooth boundary. Bollettino U.M.I. 7-A (1993), 387–395

  10. Hill, C.D., Nacinovich, M.: Pseudoconcave \(CR\) manifolds. In: Ancona, V., Ballico, E., Silva, A. (eds.) Complex Analysis and Geometry. Lecture Notes in Pure and Applied Mathematics, vol. 173, pp. 275–297. Marcel Dekker, New York (1996). (Dipartimento de Matematica, Pisa 1–76, 723 (1993)]

  11. Hill, C.D., Nacinovich, M.: On the failure of the Poincaré lemma for \(\overline{\partial }_M\) II. Math. Ann. 335, 193–219 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  12. Hill, C.D., Nacinovich, M.: Non completely solvable systems of complex first order PDE’s. Rend. Sem. Mat. Padova 129, 129–169 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  13. Hill, C.D., Nacinovich, M.: The topology of Stein CR manifolds and the Lefschetz theorem. Ann. Inst. Fourier Grenoble 43, 459–468 (1993)

    Article  MathSciNet  MATH  Google Scholar 

  14. Hörmander, Lars: Differential operators of principal type. Math. Ann. 140, 124–146 (1960)

    Article  MathSciNet  MATH  Google Scholar 

  15. Hörmander, Lars: Differential equations without solutions. Math. Ann. 140, 169–173 (1960)

    Article  MathSciNet  MATH  Google Scholar 

  16. Lewy, H.: An example of a smooth linear partial differential equation without solution. Ann. Math. (2) 66, 155–158 (1957)

    Article  MathSciNet  MATH  Google Scholar 

  17. Nacinovich, M.: On the absence of Poincaré lemma for some systems of partial differential equations. Compos. Math. 44, 241–303 (1981)

    MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Mathematisches Institut, Universität Leipzig, Augustusplatz 10/11, 04109, Leipzig, Germany

    Judith Brinkschulte

  2. Department of Mathematics, Stony Brook University, Stony Brook, NY, 11794, USA

    C. Denson Hill

  3. Dipartimento di Matematica, II Università di Roma “Tor Vergata”, Via della Ricerca Scientifica, 00133, Rome, Italy

    Mauro Nacinovich

Authors
  1. Judith Brinkschulte
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. C. Denson Hill
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Mauro Nacinovich
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Judith Brinkschulte.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Brinkschulte, J., Hill, C.D. & Nacinovich, M. On the nonvanishing of abstract Cauchy–Riemann cohomology groups. Math. Ann. 365, 1701–1715 (2016). https://doi.org/10.1007/s00208-015-1298-5

Download citation

  • Received:

  • Revised:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00208-015-1298-5

Keywords

Search

Navigation