Abstract
In this paper we study the complex basic cohomology of transversely Hermitian foliations. We use the methods developed in [7] and prove that for transversely Kähler foliations the foliated version of the Frölicher spectral sequence collapses at the first level and that the minimal model for the complex basic cohomology is formal. To stress that these properties are particular to transversely Kähler foliations we construct examples of transversely Hermitian foliations for which these theorems do not hold.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
Cordero L. A.,Holomorphic principal torus bundles, curvature and compact complex nilmanifolds, Proceedings Workshop on Curvature Geometry (C. T. J. Dodson Ed.), Lancaster Univ. (U. K.) January 1989, pp. 107–149, Lancaster Univ. 1989.
Cordero L. A., Fernández M., Gray A.,La suite spectral de Frölicher et les nilvariétés complexes compactes, C. R. Acad. Sci. Paris,305 (1987), pp. 753–756.
Cordero L. A., Fernández M., Gray A.,The Frölicher spectral sequence for compact nilmanifolds, Illinois J. Math., to appear.
Cordero L. A., Fernández M., Gray A.,Compact symplectic manifolds not admitting positive definite Kähler metrics, in Proceedings Int. Conference on Topology and its Appl., Baku, october 3–9, 1987, U. S. S. R., to appear.
Cordero L. A., Wolak R. A.,Examples of foliations with foliated geometric structures, Pacific J. Math.,142 (1990), 265–276.
Deligne P., Griffitths Ph., Morgan J., Sullivan D.,Real homotopy theory of Kähler manifolds, Invent. Math.,29 (1975), pp. 245–274.
El Kacimi-Alaoui A.,Opérateurs transversalement elliptiques sur un feuilletage riemannien et applications, Comp. Math.,73 (1990), pp. 57–106.
El Kacimi-Alaoui A., Hector G.,Décomposition de Hodge basique pour un feuilletage riemannien, Ann. Inst. Fourier,36 (1986), pp. 207–227.
Frölicher A.,Relations between the cohomology groups of Dolbeault and topological invariants, Proc. Nat. Acad. Sci. U. S. A.,41 (1955), pp. 641–644.
Griffiths Ph., Harris J.,Principles of Algebraic Geometry, Wiley Interscience 1978.
Malc’ev A. I.,A class of homogeneous spaces, Izvestia Akademii Nauk SSSR. Seriya Mathematičeskaya,13 (1949), pp. 9–32; English translation: Amer. Math. Soc. Transl.,39 (1951).
Molino P.,Riemannian Foliations, Progress in Math. vol.73, Birkhäuser 1988.
Nomizu K.,On the cohomology of compact homogeneous spaces of nilpotent Lie groups, Ann. of Math.,53 (1954), pp. 531–538.
Raghunathan M. S.,Discrete Subgroups of Lie Groups, Springer 1972.
Sergiescu V.,Cohomologie basique et dualité des feuilletages riemanniens, Ann. Inst. Fourier,35 (1985), 137–158.
Thurston W. P.,Some examples of symplectic manifolds, Proc. A. M. S.,55 (1976), pp. 467–468.
Wolak R. A.,Foliated and associated geometric structures on foliated manifolds, Ann. Fac. Sc. Toulouse X (1989), pp. 337–360.
Author information
Authors and Affiliations
Additional information
Supported by a grant of the Ministerio de Educación y Ciencia (Spain), programa «Postdoctoral Reciente».
Rights and permissions
About this article
Cite this article
Cordero, L.A., Wolak, R.A. Properties of the basic cohomology of transversely Kähler foliations. Rend. Circ. Mat. Palermo 40, 177–188 (1991). https://doi.org/10.1007/BF02844686
Received:
Accepted:
Issue Date:
DOI: https://doi.org/10.1007/BF02844686