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

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.

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.

@article{AIF_2000__50_3_995_0,
     author = {Asuke, Taro},
     title = {On the real secondary classes of transversely holomorphic foliations},
     journal = {Annales de l'Institut Fourier},
     publisher = {Association des Annales de l'institut Fourier},
     volume = {50},
     number = {3},
     year = {2000},
     pages = {995-1017},
     doi = {10.5802/aif.1782},
     zbl = {0964.58018},
     mrnumber = {2001i:58040},
     language = {en},
     url = {http://www.numdam.org/item/AIF_2000__50_3_995_0}
}
Asuke, Taro. On the real secondary classes of transversely holomorphic foliations. Annales de l'Institut Fourier, Volume 50 (2000) no. 3, pp. 995-1017. doi : 10.5802/aif.1782. http://www.numdam.org/item/AIF_2000__50_3_995_0/

[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 2000c:57052 | Zbl 0931.57021

[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 51 #14092 | Zbl 0268.57011

[6] Y. Benoist, Actions propres sur les espaces homogenes reductifs, Annals of Math., 144 (1996), 315-347. | MR 97j:22023 | Zbl 0868.22013

[7] R. Bott, On the Lefschetz Formula and Exotic Characteristic Classes, Symposia Math., 10 (1972), 95-105. | MR 50 #14778 | Zbl 0254.57013

[8] R. Bott, R. Gilter, I.M. James, Lectures on Algebraic and Differential Topology, Lecture Notes in Mathematics, No. 279, Springer-Verlag, 1972. | MR 49 #6216 | Zbl 0233.00013

[9] R. Bott, A. Haefliger, On characteristic classes of Г-foliations, Bull. Amer. Math. Soc., 78 (1972), 1039-1044. | MR 46 #6370 | Zbl 0262.57010

[10] C. Godbillon, Séminaire Bourbaki, 1972/1973, n° 421, Lecture Notes in Mathematics, No. 383, 69-87. | Numdam | Zbl 0296.17010

[11] J. Heitsch, Deformations of Secondary Characteristic Classes, Topology, 12 (1973), 381-388. | MR 47 #9639 | Zbl 0268.57010

[12] J. Heitsch, Independent variation of secondary classes, Annals of Math., 108 (1978), 421-460. | MR 80b:57022 | Zbl 0398.57007

[13] S. Hurder, Independent Rigid Secondary Classes for Holomorphic Foliations, Invent. Math., 66 (1982), 313-323. | MR 83h:57036 | Zbl 0489.57006

[14] S. Hurder and A. Katok, Ergodic theory and Weil measures for foliations, Annals of Math., 126 (1987), 221-275. | MR 89d:57042 | Zbl 0645.57021

[15] D. Husemoller, Fibre Bundles, Graduate Texts in Mathematics 20, Springer-Verlag, 1993. | Zbl 0794.55001

[16] F. W. Kamber and P. Tondeur, Foliated Bundles and Characteristic Classes, Lecture Notes in Mathematics, No. 493, Springer-Verlag, 1975. | MR 53 #6587 | Zbl 0308.57011

[17] S. Kobayashi, K. Nomizu, Foundations of Differential Geometry, Vol. II, John Wiley & Sons, Inc. | Zbl 0175.48504 | Zbl 0119.37502

[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 1193.22010 | Zbl pre05068928

[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. | MR 97j:53053 | Zbl 0852.22011

[20] S. Morita, Discontinuous Invariants of Foliations, Advanced Studies in Pure Mathematics 5, 1985, 169-193. | MR 88f:57052 | Zbl 0678.57011

[21] S. Morita, private communication.

[22] H. V. Pittie, Characteristic classes of foliations, Research Notes in Mathematics, 10, Pitman Publishing, 1976. | MR 56 #13229 | Zbl 0338.57010

[23] G. Raby, Invariance des classes de Godbillon-Vey par C1-diffeomorphisms, Ann. Inst. Fourier, Grenoble, 38-1 (1988), 205-213. | Numdam | MR 89j:57023 | Zbl 0596.57018

[24] O. H. Rasmussen, Exotic Characteristic Classes for Holomorphic Foliations, Invent. Math., 46 (1978), 153-171. | MR 80d:57015 | Zbl 0361.57017 | Zbl 0369.57007

[25] O. H. Rasmussen, Continuous Variation on Foliations in Codimension Two, Topology, 19 (1980), 335-349. | MR 82i:57027 | Zbl 0443.57021

[26] W. Thurston, Noncobordant foliations of S3, Bull. Amer. Math. Soc., 78 (1972), 511-514. | MR 45 #7741 | Zbl 0266.57004

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