Rigidity and vanishing of basic Dolbeault cohomology of Sasakian manifolds

The basic Dolbeault cohomology of a Sasakian manifold $M$ is an invariant of its characteristic foliation $\mathcal{F}$ (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 $h^{p,q} (M, \mathcal{F})$ only depend on the isomorphism class of the underlying CR structure. Equivalently, we show that they are invariant under deformations of type I. This result allows one to reduce their computation to the quasi-regular case. In the second part, we show a basic version of the Carrell-Lieberman theorem relating the basic Dolbeault cohomology of $M$ to that of the union of closed leaves of $\mathcal{F}$. As a special case, if $\mathcal{F}$ has only finitely many closed leaves, then we get $h^{p,q} (M, \mathcal{F})$ for $p \neq q$. Combining the two results, we obtain the same vanishing result if $M$ admits a nowhere vanishing CR vector field with finitely many closed orbits. As an application of these results, we compute the basic Hodge numbers for toric Sasakian manifolds and deformations of homogeneous Sasakian manifolds.

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Published 24 June 2016