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.
Highlights
In this short paper, we study certain properties of deformations of transversely holomorphic foliations
We give a positive answer to the question, i.e. we prove the following theorem: Theorem 1.1 Given a smooth family {(Ms, s, s, gs, s)}s∈[0,1] of compact Sasakian manifolds and fixed integers p and q the function associating to each point s ∈ [0, 1] the basic Hodge number hps,q of (Ms, s, s, gs, s) is constant
First we prove Theorem 3.1 which states that the basic Hodge numbers are constant for any smooth family of manifolds with homologically orientable transverse Kähler foliations for which the spaces of complex-valued basic harmonic forms constitute a bundle over the interval
Summary
We study certain properties of deformations of transversely holomorphic foliations. On the way we correct a slight inaccuracy in [13] (see Remark 3.3) This theorem strongly relies on the Sasaki structure (and on the transverse Kähler structure) and so the following question remains open: Question 1.2 Are the basic Hodge numbers rigid under deformations of (homologically orientable) transversely Kähler foliations on compact manifolds?. We show that if the basic ∂∂̄-lemma holds for a foliated manifold (M, F) , it holds for appropriately small deformations of the transverse holomorphic structure (provided that we do not deform the foliation itself) as well as a similar rigidity theorem for being transversely Kähler These results aside from the upper semi-continuity theorem for the Bott–Chern and Aeppli cohomology use the Frölicher-type inequality for foliations which was proven in [19].
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