On the real secondary classes of transversely holomorphic foliations

Asuke, Taro

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 $H^{*}({\mathrm{WO}}_{2q})$ des classes secondaires réelles vers l’espace $H^{*}({\mathrm{WU}}_q)$ 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 $H^{*}({\mathrm{WO}}_{2q})$ of the real secondary classes to the space $H^{*}({\mathrm{WU}}_q)$ 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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