Rigidity and vanishing of basic Dolbeault cohomology of Sasakian manifolds

TL;DR

This paper investigates the properties of basic Dolbeault cohomology in Sasakian manifolds, showing invariance under certain deformations and relating it to the structure of closed leaves, with applications to homogeneous cases.

Contribution

It establishes invariance of basic Hodge numbers under deformation and relates cohomology to closed leaves, simplifying computations for Sasakian manifolds.

Findings

Basic Hodge numbers depend only on the CR structure.

Hodge numbers vanish when the foliation has finitely many closed leaves.

Computed Hodge numbers for deformations of homogeneous Sasakian manifolds.

Abstract

The basic Dolbeault cohomology groups of a Sasakian manifold M are invariants 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, the dimensions of the basic Dolbeault cohomology groups, 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 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,F) to that of (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 the (p,q)-th basic Hodge numbers are zero for p not equal to q. Combining the two results, we show that if M admits a nowhere vanishing CR vector field with finitely many closed orbits, then the (p,q)-th basic Hodge numbers are zero for p not equal to q. As an application of these results, we compute then the basic Hodge numbers for deformations of homogeneous Sasakian manifolds.