Invariance of basic Hodge numbers under deformations of Sasakian manifolds

TL;DR

This paper proves the invariance of basic Hodge numbers under deformations of Sasakian manifolds and explores the rigidity of certain geometric properties, with implications for the stability of transversely Kähler structures.

Contribution

It establishes the invariance of Hodge numbers and the rigidity of the $ar{ ext{d}}$-lemma and transversely Kähler property under small deformations of Sasakian structures.

Findings

Hodge numbers are invariant under deformations.

The $ar{ ext{d}}$-lemma and transversely Kähler property are rigid under small deformations.

Counterexamples show these properties are not preserved under arbitrary deformations.

Abstract

We show that the Hodge numbers of Sasakian manifolds are invariant under arbitrary deformations of the Sasakian structure. We also present an upper semi continuity Theorem for the dimensions of kernels of a smooth family of transversely elliptic operators on manifolds with transversely Riemannian foliations. We use this to prove that the $\partial\bar{\partial}$-lemma and being transversely K\"{a}hler are rigid properties under small deformations of the transversely holomorphic structure which preserve the foliation. Finally, we study an example which shows that this is not the case for arbitrary deformations of the transversely holomorphic foliation.