Remote Access Transactions of the American Mathematical Society
Green Open Access

Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 

 

Complex-foliated structures. I. Cohomology of the Dolbeault-Kostant complexes


Authors: Hans R. Fischer and Floyd L. Williams
Journal: Trans. Amer. Math. Soc. 252 (1979), 163-195
MSC: Primary 58A30; Secondary 32L10, 58F06, 81C40
MathSciNet review: 534116
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: We study the cohomology of differential complexes, which we shall call Dolbeault-Kostant complexes, defined by certain integrable sub-bundles F of the complex tangent bundle of a manifold M. When M has a complex or symplectic structure and F is chosen to be the bundle of anti-holomorphic tangent vectors or, respectively, a ``polarization'' then the corresponding complexes are, respectively, the Dolbeault complex and (under further conditions) a complex introduced by Kostant in the context of geometric quantization. A simple condition on F insures that our complexes are elliptic. Assuming ellipticity and compactness of M, for example, one of our key results is a Hirzebruch-Riemann-Roch Theorem.


References [Enhancements On Off] (What's this?)

  • [1] A. Borel and F. Hirzebruch, Characteristic classes and homogeneous spaces. I, Amer. J. Math. 80 (1958), 458–538. MR 0102800
  • [2] N. Bourbaki, Éléments de mathématique. Fasc. XXXVI. Variétés différentielles et analytiques. Fascicule de résultats (Paragraphes 8 à 15), Actualités Scientifiques et Industrielles, No. 1347, Hermann, Paris, 1971 (French). MR 0281115
  • [3] Pierre Dolbeault, Formes différentielles et cohomologie sur une variété analytique complexe. I, Ann. of Math. (2) 64 (1956), 83–130 (French). MR 0083166
  • [4] R. E. Edwards, Functional analysis. Theory and applications, Holt, Rinehart and Winston, New York-Toronto-London, 1965. MR 0221256
  • [5] Alfred Frölicher, Relations between the cohomology groups of Dolbeault and topological invariants, Proc. Nat. Acad. Sci. U.S.A. 41 (1955), 641–644. MR 0073262
  • [6] Werner Greub, Stephen Halperin, and Ray Vanstone, Connections, curvature, and cohomology, Academic Press [Harcourt Brace Jovanovich, Publishers], New York-London, 1976. Volume III: Cohomology of principal bundles and homogeneous spaces; Pure and Applied Mathematics, Vol. 47-III. MR 0400275
  • [7] Alexander Grothendieck, Sur quelques points d’algèbre homologique, Tôhoku Math. J. (2) 9 (1957), 119–221 (French). MR 0102537
  • [8] F. Hirzebruch, Topological methods in algebraic geometry, Third enlarged edition. New appendix and translation from the second German edition by R. L. E. Schwarzenberger, with an additional section by A. Borel. Die Grundlehren der Mathematischen Wissenschaften, Band 131, Springer-Verlag New York, Inc., New York, 1966. MR 0202713
  • [9] A. A. Kirillov, Elements of the theory of representations, Springer-Verlag, Berlin-New York, 1976. Translated from the Russian by Edwin Hewitt; Grundlehren der Mathematischen Wissenschaften, Band 220. MR 0412321
  • [10] B. Kostant, MIT Seminar, 1967, also in: Lectures in modern analysis and applications. III, Lecture Notes in Math., vol. 170, Springer-Verlag, Berlin, 1970.
  • [11] Serge Lang, Differential manifolds, Addison-Wesley Publishing Co., Inc., Reading, Mass.-London-Don Mills, Ont., 1972. MR 0431240
  • [12] L. Nirenberg, A complex Frobenius theorem, Seminars on Analytic Functions. I, Princeton Univ. Press, Princeton, N. J., 1957.
  • [13] Richard S. Palais, Seminar on the Atiyah-Singer index theorem, With contributions by M. F. Atiyah, A. Borel, E. E. Floyd, R. T. Seeley, W. Shih and R. Solovay. Annals of Mathematics Studies, No. 57, Princeton University Press, Princeton, N.J., 1965. MR 0198494
  • [14] J. H. Rawnsley, On the cohomology groups of a polarisation and diagonal quantisation, Trans. Amer. Math. Soc. 230 (1977), 235–255. MR 0648775, 10.1090/S0002-9947-1977-0648775-2
  • [15] P. Renouard, Variétés symplectiques et quantification, thèse, Orsay, 1969.
  • [16] G. de Rham, Variétés différentiables, Hermann, Paris, 1955.
  • [17] K. S. Sarkaria, The de Rham cohomology of foliated manifolds, thesis, SUNY, Stony Brook, 1974.
  • [18] Jean-Pierre Serre, Un théorème de dualité, Comment. Math. Helv. 29 (1955), 9–26 (French). MR 0067489
  • [19] François Trèves, Topological vector spaces, distributions and kernels, Academic Press, New York-London, 1967. MR 0225131
  • [20] Izu Vaisman, Cohomology and differential forms, Marcel Dekker, Inc., New York, 1973. Translation editor: Samuel I. Goldberg; Pure and Applied Mathematics, 21. MR 0341344
  • [21] André Weil, Introduction à l’étude des variétés kählériennes, Publications de l’Institut de Mathématique de l’Université de Nancago, VI. Actualités Sci. Ind. no. 1267, Hermann, Paris, 1958 (French). MR 0111056
  • [22] R. O. Wells Jr., Differential analysis on complex manifolds, Prentice-Hall, Inc., Englewood Cliffs, N.J., 1973. Prentice-Hall Series in Modern Analysis. MR 0515872

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC: 58A30, 32L10, 58F06, 81C40

Retrieve articles in all journals with MSC: 58A30, 32L10, 58F06, 81C40


Additional Information

DOI: http://dx.doi.org/10.1090/S0002-9947-1979-0534116-X
Keywords: Sheaf cohomology, Chern class, Todd class, spectral sequence, elliptic complex
Article copyright: © Copyright 1979 American Mathematical Society