The canonical foliations of a locally conformal Kähler manifold

Abstract

Let M be a locally conformal Kahler manifold. Then the Kahler form Ω of M satisfies dΩ=ωΩ for some closed 1 -form ω, called the Lee form of M. We show that M admits three canonical foliations (four if ω is parallel) and we prove several properties of them, improving previous results of I. Vaisman. In particular all of these foliations are totally geodesic and Riemannian, and one of them is also almost complex. If this latter foliation is regular on a compact M, then we prove that M is a locally trivial fiber bundle over a compact Kahler manifold M, and the fibers are totally geodesic flat 2-tori. Finally we study geometrical properties, the canonical class and the Godbillon-Vey class of the totally real foliation of a CR-submanifold N ⊂cM.