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.
", pages = "377-399(23)", url = "http://www.ingentaconnect.com/content/klu/agag/2002/00000021/00000004/00397905" }