Invariance of basic Hodge numbers under deformations of Sasakian manifolds

Annali di Matematica Pura ed Applicata (1923 -) (2021) 200:1451–1468 https://doi.org/10.1007/s10231-020-01044-8 Invariance of basic Hodge numbers under deformations of Sasakian manifolds Paweł Raźny1 Received: 27 July 2020 / Accepted: 22 September 2020 / Published online: 7 October 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 1 Institute of Mathematics, Faculty of Mathematics and Computer Science, Jagiellonian University in Cracow, Cracow, Poland 13 Vol.:(0123456789) 1452 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 difference being that our focus is on the standard Laplace operator and not the Dolbeault-Laplace operator) and then describe basic forms as the kernel of the Lie derivative. On the way we correct a slight inaccuracy in [13] (see Remark 3.3). This theorem strongly relies on the Sasaki structure (and not only 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 feel that Theorem 3.1 might be helpful in solving this more general problem. Moreover, an answer to this question would have some further use to the theory of S-structures which were developed in [3] and are the higher-dimensional (meaning the dimension of the characteristic foliation) analogue of Sasakian structures. In Sect. 4 we develop some of the Theorems from [16] for smooth families of transversely elliptic operators on manifolds with TP foliations. We apply them to prove the upper semi-continuity Theorem of the dimensions of kernels of such operators. The key difficulty here is finding the way to bypass the TP condition required in previous theorems in this section (which can be bypassed by using the frame bundle construction). This in turn is applied to achieve our results in Sects. 5 and 7. We devote the fifth and sixth section to the study of the behaviour of the basic 𝜕 𝜕̄ -lemma under deformations of transversely holomorphic foliations. We show that if the basic 𝜕 𝜕̄-lemma holds for a foliated manifold (M, F) , then it also 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]. In Sect. 6 we show that the restriction on deforming the foliation is necessary by studying an example from [13, 15]. The final section of this paper treats the applications of the results from Sect. 4 to the transverse symplectic setting. The most notable consequence is the rigidity of the K-contact hard Lefschetz property under deformations which preserve the Reeb foliation. Due to the fact that here we leave the transversely holomorphic setting we try to make this section as much self-contained as possible. 13 Invariance of basic Hodge numbers under deformations of Sasakian… 1453 2 Preliminaries 2.1 Foliations We provide a quick review of transverse structures on foliations. Definition 2.1 A codimension q foliation F on a smooth n-manifold M is given by the fol- lowing data: • An open cover U ∶= {Ui }i∈I of M. • A q-dimensional smooth manifold T0. • For each Ui ∈ U a submersion fi ∶ Ui → T0 with connected fibers (these fibers are called plaques). • For all intersections Ui ∩ Uj ≠ � a local diffeomorphism 𝛾ij of T0 such that fj = 𝛾ij ◦fi The last condition ensures that plaques glue nicely to form a partition of M consisting of submanifolds of M of codimension q. This partition is called a foliation F of M and the elements of this partition are called leaves of F . ∐ fi (Ui ) the transverse manifold of F . The local diffeomorphisms 𝛾ij genWe call T = Ui ∈U erate a pseudogroup Γ of transformations on T (called the holonomy pseudogroup).The space of leaves M∕F of the foliation F can be identified with T∕Γ. Definition 2.2 A smooth form 𝜔 on M is called basic if for any vector field X tangent to the leaves of F the following equality holds: iX 𝜔 = iX d𝜔 = 0. Basic 0-forms will be called basic functions henceforth. Basic forms are in one to one correspondence with Γ-invariant smooth forms on T. (...truncated)