Real Polarizable Hodge Structures Arising from Foliations

Authors: Deninger C.¹; Singhof W.²

Source: Annals of Global Analysis and Geometry, Volume 21, Number 4, June 2002 , pp. 377-399(23)

Publisher: Springer

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Abstract:

We construct real polarizable Hodge structures on the reduced leafwise cohomology of Kähler–Riemann foliations by complex manifolds. As in the classical case one obtains a hard Lefschetz theorem for this cohomology. Serre's Kählerin analogue of the Weil conjectures carries over as well. Generalizing a construction of Looijenga and Lunts one obtains possibly infinite-dimensional Lie algebras attached to Kähler–Riemann foliations.

Finally using (\mathfrak g, K)-cohomology we discuss a class of examples obtained by dividing a product of symmetric spaces by a cocompact lattice and considering the foliations coming from the factors.

Keywords: foliation; leafwise cohomology; Hodge structure; symmetric spaces

Language: English

Document Type: Regular paper

Affiliations: 1: Mathematisches Institut, WWU Münster, Einsteinstr. 62, 48149 Münster, Germany. e-mail: deninge@math.uni-muenster.de 2: Mathematisches Institut, Universität Düsseldorf, Universitätsstr. 1, 40225 Düsseldorf, Germany. e-mail: singhof@cs.uni-duesseldorf.de

Publication date: 2002-06-01