The Journal of Geometric Analysis

, Volume 1, Issue 1, pp 39–70 | Cite as

Homology and cohomology in hypo-analytic structures of the hypersurface type

  • Paulo Cordaro
  • François Treves


The work is concerned with local exactness in the cohomology of the differential complex associated with a hypo-analytic structure on a smooth manifold. Only structures of the hypersurface type are considered, i.e., structures in which the rank of the characteristic set does not exceed one. Among them are the CR structures of real hypersurfaces in a complex manifold. The main theorem states anecessary condition for local exactness in dimensionq to hold. The condition is stated in terms of the natural homology associated with the differential complex, as inherited by the level sets of the imaginary part of an arbitrary solutionw whose differential spans the characteristic set at the central point. An intersection number, which generalizes the standard number in singular homology, is defined; the condition is that this number, applied to the intersection of the level sets ofImw with the hypersurfaceRew=0, vanish identically. In a CR structure, and in top dimension, this is shown to be equivalent to the property that the Levi form not be definite at any point—a property, that is likely to be also sufficient for local solvability.

AMS Subject Classification

No. 39 

Key Words and Phrases

Cohomology current differential form homology locally integrable system of vector fields local solvability sheaf 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. [1]
    Andreotti, A., and Hill, C. D. E. E. Levi convexity and the Hans Lewy problem I. Ann. Scuola Norm. Sup. Pisa Sci. Fis. Mat.26, 325–363 (1972). E. E. Levi convexity and the Hans Lewy problem II. Ann. Scuola Norm. Sup. Pisa Sci. Fis. Mat.26, 747–806 (1972).MATHMathSciNetGoogle Scholar
  2. [2]
    Baouendi, M. S., Chang C., and Treves, F. Microlocal analyticity and extension of CR functions. J. Diff. Geometry18, 331–391 (1983).MATHMathSciNetGoogle Scholar
  3. [3]
    Baouendi, M. S., and Treves, F. A property of the functions and distributions annihilated by a locally integrable system of complex vector fields. Ann. Math.113, 387–421 (1981).CrossRefMathSciNetGoogle Scholar
  4. [4]
    Cartan, H. Séminaires E.N.S. 1951/52.Google Scholar
  5. [5]
    Cordaro, P., and Hounie, J. On local solvability of underdetermined systems of vector fields. Am. J. Math.112, 243–270 (1990).MATHCrossRefMathSciNetGoogle Scholar
  6. [6]
    De Rham, G. Varietés différentiables. Paris 1955.Google Scholar
  7. [7]
    Godement, R. Topologie Algébrique et Théorie des Faisceaux. Paris: Hermann 1958.MATHGoogle Scholar
  8. [8]
    Hörmander, L. Differential equations without solution. Math. Ann.140, 169–173 (1960).MATHCrossRefMathSciNetGoogle Scholar
  9. [9]
    Mendoza, G., and Treves, F. Local solvability in a class of overdetermined systems of linear PDE. Duke Math. J. (1991). To appear.Google Scholar
  10. [10]
    Nirenberg, L., and Treves, F. Solvability of a first-order partial differential equation. Comm. Pure Appl. Math.16, 331–351 (1963).MATHCrossRefMathSciNetGoogle Scholar
  11. [11]
    Schwartz, L. Théorie des distribution. 2nd ed. Paris: Hermann 1966.Google Scholar
  12. [12]
    Treves, F. Study of a model in the theory of complexes of pseudodifferential operators. Ann. Math.104, 269–324 (1976).CrossRefMathSciNetGoogle Scholar
  13. [13]
    Treves, F. On the local integrability and local solvability of systems of vector fields. Acta Math.151, 1–48 (1983).MATHCrossRefMathSciNetGoogle Scholar
  14. [14]
    Treves, F. On the local solvability for top degree forms in hypo-analytic structures. Am. J. Math.,112, 403–421 (1990).MATHCrossRefMathSciNetGoogle Scholar
  15. [15]
    Treves, F. Hypo-analytic structures. Princeton, NJ: Princeton University Press 1991.Google Scholar

Copyright information

© Mathematica Josephina, Inc. 1991

Authors and Affiliations

  • Paulo Cordaro
    • 1
    • 2
  • François Treves
    • 1
    • 2
  1. 1.Instituto de Matemática e EstatistícaUniversidade de São PauloSão PauloBrazil
  2. 2.Mathematics DepartmentRutgers UniversityNew BrunswickUSA

Personalised recommendations