In this paper we study the real secondary classes of transversely holomorphic foliations. We define a homomorphism from the space of the real secondary classes to the space 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 des classes secondaires réelles vers l’espace 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}, pages = {995--1017}, publisher = {Association des Annales de l{\textquoteright}institut Fourier}, volume = {50}, number = {3}, year = {2000}, doi = {10.5802/aif.1782}, mrnumber = {2001i:58040}, zbl = {0964.58018}, language = {en}, url = {http://archive.numdam.org/articles/10.5802/aif.1782/} }
TY - JOUR AU - Asuke, Taro TI - On the real secondary classes of transversely holomorphic foliations JO - Annales de l'Institut Fourier PY - 2000 SP - 995 EP - 1017 VL - 50 IS - 3 PB - Association des Annales de l’institut Fourier UR - http://archive.numdam.org/articles/10.5802/aif.1782/ DO - 10.5802/aif.1782 LA - en ID - AIF_2000__50_3_995_0 ER -
%0 Journal Article %A Asuke, Taro %T On the real secondary classes of transversely holomorphic foliations %J Annales de l'Institut Fourier %D 2000 %P 995-1017 %V 50 %N 3 %I Association des Annales de l’institut Fourier %U http://archive.numdam.org/articles/10.5802/aif.1782/ %R 10.5802/aif.1782 %G en %F 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://archive.numdam.org/articles/10.5802/aif.1782/
[1] Invariance of the Godbillon-Vey class by C1-diffeomorphisms for higher codimensional foliations, Jour. Math. Soc. Japan, 51 (1999), 655-660. | MR | Zbl
,[2] On the real secondary classes of transversely holomorphic foliations, University of Tokyo, Thesis.
,[3] On the real secondary classes of transversely holomorphic foliations II, preprint.
,[4] A remark on the Bott class, preprint.
,[5] Singularities of Holomorphic Foliations, Jour. Diff. Geom., 7 (1972), 279-342. | MR | Zbl
and ,[6] Actions propres sur les espaces homogenes reductifs, Annals of Math., 144 (1996), 315-347. | MR | Zbl
,[7] On the Lefschetz Formula and Exotic Characteristic Classes, Symposia Math., 10 (1972), 95-105. | MR | Zbl
,[8] Lectures on Algebraic and Differential Topology, Lecture Notes in Mathematics, No. 279, Springer-Verlag, 1972. | MR | Zbl
, , ,[9] On characteristic classes of Г-foliations, Bull. Amer. Math. Soc., 78 (1972), 1039-1044. | MR | Zbl
, ,[10] Séminaire Bourbaki, 1972/1973, n° 421, Lecture Notes in Mathematics, No. 383, 69-87. | EuDML | Numdam | Zbl
,[11] Deformations of Secondary Characteristic Classes, Topology, 12 (1973), 381-388. | MR | Zbl
,[12] Independent variation of secondary classes, Annals of Math., 108 (1978), 421-460. | MR | Zbl
,[13] Independent Rigid Secondary Classes for Holomorphic Foliations, Invent. Math., 66 (1982), 313-323. | EuDML | MR | Zbl
,[14] Ergodic theory and Weil measures for foliations, Annals of Math., 126 (1987), 221-275. | MR | Zbl
and ,[15] Fibre Bundles, Graduate Texts in Mathematics 20, Springer-Verlag, 1993. | Zbl
,[16] Foliated Bundles and Characteristic Classes, Lecture Notes in Mathematics, No. 493, Springer-Verlag, 1975. | MR | Zbl
and ,[17] Foundations of Differential Geometry, Vol. II, John Wiley & Sons, Inc. | Zbl
, ,[18] Discontinuous Groups Acting on Homogeneous Spaces of Reductive Type, Representation Theory of Lie Groups and Lie Algebras, World Scientific, 1992, 59-75. | Zbl
,[19] On manifolds locally modelled on non-Riemannian homogeneous spaces, Geom. Funct. Anal., 5 (1995), 955-965. | EuDML | MR | Zbl
, , and ,[20] Discontinuous Invariants of Foliations, Advanced Studies in Pure Mathematics 5, 1985, 169-193. | MR | Zbl
,[21] private communication.
,[22] Characteristic classes of foliations, Research Notes in Mathematics, 10, Pitman Publishing, 1976. | MR | Zbl
,[23] Invariance des classes de Godbillon-Vey par C1-diffeomorphisms, Ann. Inst. Fourier, Grenoble, 38-1 (1988), 205-213. | EuDML | Numdam | MR | Zbl
,[24] Exotic Characteristic Classes for Holomorphic Foliations, Invent. Math., 46 (1978), 153-171. | EuDML | MR | Zbl
,[25] Continuous Variation on Foliations in Codimension Two, Topology, 19 (1980), 335-349. | MR | Zbl
,[26] Noncobordant foliations of S3, Bull. Amer. Math. Soc., 78 (1972), 511-514. | MR | Zbl
,[27] The Godbillon-Vey class of transversely holomorphic foliations, preprint.
,Cited by Sources: