Invariance of basic Hodge numbers under deformations of Sasakian manifolds
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Invariance of basic Hodge numbers under deformations of Sasakian manifolds Paweł Raźny1 Received: 27 July 2020 / Accepted: 22 September 2020 © The Author(s) 2020
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 𝜕 𝜕̄-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. Keywords Sasakian manifolds · Foliations · Basic cohomology Mathematics Subject Classification 53C12 · 53C25
1 Introduction In this short paper, we study certain properties of deformations of transversely holomorphic foliations. In [13] the authors pose the question whether the basic Hodge numbers of Sasakian manifolds are rigid under arbitrary deformations of Sasakian manifolds. This is motivated by their results on the invariance of such numbers under type I and type II deformations as well as the fact that basic Hodge numbers can be used to distinguish different Sasaki structures on a given manifold. We give a positive answer to the question, i.e. we prove the following theorem:
Theorem 1.1 Given a smooth family {(Ms , 𝜉s , 𝜂s , gs , 𝜙s )}s∈[0,1] of compact Sasakian manifolds and fixed integers p and q the function associating to each point s ∈ [0, 1] the basic p,q Hodge number hs of (Ms , 𝜉s , 𝜂s , gs , 𝜙s ) is constant.
* Paweł Raźny [email protected] 1
Institute of Mathematics, Faculty of Mathematics and Computer Science, Jagiellonian University in Cracow, Cracow, Poland
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P. Raźny
We split the proof of this result into two theorems which are of independent interest. First we prove Theorem 3.1 which states that the basic Hodge numbers are constant for any smooth family (over the interval [0, 1]) of manifolds with homologically orientable transverse Kähler foliations for which the spaces of complex-valued basic harmonic forms constitute a bundle over the interval. Since a family of Sasakian manifolds is in particular a family of homologically orientable transversely Kähler foliations all that is left to prove is that in this case the spaces of complex-valued basic harmonic forms give in fact a bundle over the interval. This is precisely the content of Theorem 3.4 which allows us to bypass the key difficulty of this and related problems (such as in [13]) meaning the fact that the spaces of basic forms over each manifold do not in general form a bundle over the interval. The idea of the proof of this theorem is to first treat transverse forms following [13] (the differe
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