On the real secondary classes of transversely holomorphic foliations
Annales de l'Institut Fourier, Tome 50 (2000) no. 3, pp. 995-1017.

Dans cet article nous étudions les classes caractéristiques secondaires réelles de feuilletages transversalement holomorphes. Nous définissons un homomorphisme de l’espace H * ( WO 2q ) des classes secondaires réelles vers l’espace H * ( WU q ) des classes secondaires complexes qui correspond à oublier la structure transversalement holomorphe. En utilisant cet homomorphisme nous montrons, par exemple, la décomposition de la classe de Godbillon-Vey en la partie imaginaire de la classe de Bott et la première classe de Chern du fibré normal complexe du feuilletage. Nous montrons aussi que des exemples de Heitsch n’admettent pas de structure transversalement holomorphe.

In this paper we study the real secondary classes of transversely holomorphic foliations. We define a homomorphism from the space H * ( WO 2q ) of the real secondary classes to the space H * ( WU q ) of the complex secondary classes that corresponds to forgetting the transverse holomorphic structure. By using this homomorphism we show, for example, the decomposition of the Godbillon-Vey class into the imaginary part of the Bott class and the first Chern class of the complex normal bundle of the foliation. We show also that Heitsch’s examples do not admit any transverse holomorphic structure.

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Asuke, Taro. On the real secondary classes of transversely holomorphic foliations. Annales de l'Institut Fourier, Tome 50 (2000) no. 3, pp. 995-1017. doi : 10.5802/aif.1782. http://www.numdam.org/articles/10.5802/aif.1782/

[1] T. Asuke, Invariance of the Godbillon-Vey class by C1-diffeomorphisms for higher codimensional foliations, Jour. Math. Soc. Japan, 51 (1999), 655-660. | MR | Zbl

[2] T. Asuke, On the real secondary classes of transversely holomorphic foliations, University of Tokyo, Thesis.

[3] T. Asuke, On the real secondary classes of transversely holomorphic foliations II, preprint.

[4] T. Asuke, A remark on the Bott class, preprint.

[5] P. Baum and R. Bott, Singularities of Holomorphic Foliations, Jour. Diff. Geom., 7 (1972), 279-342. | MR | Zbl

[6] Y. Benoist, Actions propres sur les espaces homogenes reductifs, Annals of Math., 144 (1996), 315-347. | MR | Zbl

[7] R. Bott, On the Lefschetz Formula and Exotic Characteristic Classes, Symposia Math., 10 (1972), 95-105. | MR | Zbl

[8] R. Bott, R. Gilter, I.M. James, Lectures on Algebraic and Differential Topology, Lecture Notes in Mathematics, No. 279, Springer-Verlag, 1972. | MR | Zbl

[9] R. Bott, A. Haefliger, On characteristic classes of Г-foliations, Bull. Amer. Math. Soc., 78 (1972), 1039-1044. | MR | Zbl

[10] C. Godbillon, Séminaire Bourbaki, 1972/1973, n° 421, Lecture Notes in Mathematics, No. 383, 69-87. | EuDML | Numdam | Zbl

[11] J. Heitsch, Deformations of Secondary Characteristic Classes, Topology, 12 (1973), 381-388. | MR | Zbl

[12] J. Heitsch, Independent variation of secondary classes, Annals of Math., 108 (1978), 421-460. | MR | Zbl

[13] S. Hurder, Independent Rigid Secondary Classes for Holomorphic Foliations, Invent. Math., 66 (1982), 313-323. | EuDML | MR | Zbl

[14] S. Hurder and A. Katok, Ergodic theory and Weil measures for foliations, Annals of Math., 126 (1987), 221-275. | MR | Zbl

[15] D. Husemoller, Fibre Bundles, Graduate Texts in Mathematics 20, Springer-Verlag, 1993. | Zbl

[16] F. W. Kamber and P. Tondeur, Foliated Bundles and Characteristic Classes, Lecture Notes in Mathematics, No. 493, Springer-Verlag, 1975. | MR | Zbl

[17] S. Kobayashi, K. Nomizu, Foundations of Differential Geometry, Vol. II, John Wiley & Sons, Inc. | Zbl

[18] T. Kobayashi, Discontinuous Groups Acting on Homogeneous Spaces of Reductive Type, Representation Theory of Lie Groups and Lie Algebras, World Scientific, 1992, 59-75. | Zbl

[19] F. Labourie, S. Mozes, and R. J. Zimmer, On manifolds locally modelled on non-Riemannian homogeneous spaces, Geom. Funct. Anal., 5 (1995), 955-965. | EuDML | MR | Zbl

[20] S. Morita, Discontinuous Invariants of Foliations, Advanced Studies in Pure Mathematics 5, 1985, 169-193. | MR | Zbl

[21] S. Morita, private communication.

[22] H. V. Pittie, Characteristic classes of foliations, Research Notes in Mathematics, 10, Pitman Publishing, 1976. | MR | Zbl

[23] G. Raby, Invariance des classes de Godbillon-Vey par C1-diffeomorphisms, Ann. Inst. Fourier, Grenoble, 38-1 (1988), 205-213. | EuDML | Numdam | MR | Zbl

[24] O. H. Rasmussen, Exotic Characteristic Classes for Holomorphic Foliations, Invent. Math., 46 (1978), 153-171. | EuDML | MR | Zbl

[25] O. H. Rasmussen, Continuous Variation on Foliations in Codimension Two, Topology, 19 (1980), 335-349. | MR | Zbl

[26] W. Thurston, Noncobordant foliations of S3, Bull. Amer. Math. Soc., 78 (1972), 511-514. | MR | Zbl

[27] T. Asuke, The Godbillon-Vey class of transversely holomorphic foliations, preprint.

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