The basic Dolbeault cohomology H p,q (M, F ) of a Sasakian manifold (M, η, g) is an invariant of its characteristic foliation F (the orbit foliation of the Reeb flow). We show some fundamental properties of this cohomology, which are useful for its computation. In the first part of the article, we show that the basic Hodge numbers h p,q (M, F ), the dimensions of H p,q (M, F ), only depend on the isomorphism class of the underlying CR structure. Equivalently, we show that the basic Hodge numbers are invariant under deformations of type I. This result reduces the computation of h p,q (M, F ) to the quasi-regular case. In the second part, we show a basic version of the Carrell-Lieberman theorem relating H •,• (M, F ) to H •,• (C, F ), where C is the union of closed leaves of F . As a special case, if F has only finitely many closed leaves, then we get h p,q (M, F ) = 0 for p = q. Combining the two results, we show that if M admits a nowhere vanishing CR vector field with finitely many closed orbits, then h p,q (M, F ) = 0 for p = q. As an application of these results, we compute h p,q (M, F ) for deformations of homogeneous Sasakian manifolds.2010 Mathematics Subject Classification. 53D35, 55N25, 58A14.

Abstract. Deformations of the Reeb flow of a Sasakian manifold as transversely Kähler flows may not admit compatible Sasakian metrics. We show that the triviality of the (0, 2)-component of the basic Euler class characterizes the existence of compatible Sasakian metrics for given small deformations of the Reeb flow as transversely holomorphic Riemannian flows. We also prove a Kodaira-Akizuki-Nakano type vanishing theorem for basic Dolbeault cohomology of homologically orientable transversely Kähler foliations. As a consequence of these results, we show that any small deformations of the Reeb flow of a positive Sasakian manifold admit compatible Sasakian metrics.

We investigate the equivariant cohomology of the natural torus action on a K-contact manifold and its relation to the topology of the Reeb flow. Using the contact moment map, we show that the equivariant cohomology of this action is Cohen-Macaulay, the natural substitute of equivariant formality for torus actions without fixed points. As a consequence, generic components of the contact moment map are perfect Morse-Bott functions for the basic cohomology of the orbit foliation F of the Reeb flow. Assuming that the closed Reeb orbits are isolated, we show that the basic cohomology of F vanishes in odd degrees, and that its dimension equals the number of closed Reeb orbits. We characterize K-contact manifolds with minimal number of closed Reeb orbits as real cohomology spheres. We also prove a GKM-type theorem for K-contact manifolds which allows to calculate the equivariant cohomology algebra under the nonisolated GKM condition.2010 Mathematics Subject Classification. 53D35, 53D20, 55N25.

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