Invariance of basic Hodge numbers under deformations of Sasakian manifolds

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 homologically orientable transversely Riemannian foliations. We use this to prove that the $$\partial {\bar{\partial }}$$ ∂ ∂ ¯ -lemma and being transversely Kähler are rigid properties under small deformations of the transversely holomorphic structure which preserve the foliation. We study an example which shows that this is not the case for arbitrary deformations of the transversely holomorphic foliation. Finally we point out an application of the upper-semi continuity theorem to K-contact manifolds.

On extension of closed complex (basic) differential forms: Hodge numbers and (transversely) $p$-Kähler structures

“…If F = F 0 is transversely p-Kähler and satisfies the (p, p + 1)-th mild ∂ ∂-lemma for 1 ≤ p ≤ r, then F t is also transversely p-Kähler for every t sufficiently close to 0. This result strengthens the works by [EKAG97,Rź21] and can be viewed as a generalization of [RWZ19, Theorem 1.1] to the foliated case. Notice that compared with the previous works, Theorem 1.7 uses weaker assumptions when p = 1 (transversely Kähler), see Remark 6.28.…”

“…Then F t has a transversely Kähler metric σ t for every t sufficiently close to 0, depending differentiably on t with σ 0 = σ. See also [Rź21,Theorem 5.2]. One can refer to Subsections 6.1-6.4 for relevant definitions.…”

“…Acknowledgement: Both authors would like to thank Professors A. El Kacimi Alaoui, M. Nicolau and P. Raźny, for explaining us the details on their articles [EKAN89] and [Rź21], respectively. Furthermore, we also thank Professors W. Xia and S. Zhu for giving many useful suggestions on this paper.…”

Sasaki structures distinguished by their basic Hodge numbers

“…Very recently, Raźny [28] proved that the basic Hodge numbers of Sasaki manifolds are constant in arbitrary smooth deformations of the Sasaki structure. This completes earlier results of Boyer and Galicki [4] and of Goertsches, Nozawa and Töben [15] who proved the invariance for special types of deformations.…”

“…Note that the deformations of Sasaki structures considered by Raźny [28] preserve the isotopy class of the underlying contact structure. It is not clear how the basic Hodge numbers of a Sasaki structure relate to the underlying contact structure.…”