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Real polarizable Hodge structures arising from foliations

Christopher Deninger·Wilhelm Singhof
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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 Kählerian 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.

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Christopher Deninger, Wilhelm Singhof. 2002-04-10. Real polarizable Hodge structures arising from foliations. https://arxiv.org/abs/math/0204111

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