The Frölicher-type inequalities of foliations

2020

Annali di Matematica

Self Cite

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.

Section: Introductionmentioning

confidence: 94%

“…In this subsection we provide some of the results from [19] which will be used in this paper. Let M be a manifold of dimension n = p + 2q , endowed with a Hermitian foliation F of complex codimension q.…”

Section: Basic Bott-chern and Aeppli Cohomology Theoriesmentioning

confidence: 99%

Invariance of basic Hodge numbers under deformations of Sasakian manifolds

2020

Annali di Matematica

Self Cite

“…These cohomologies (and especially their non-foliated counterparts) are subject of extensive studies (cf. [2][3][4]9,12,14]); in particular, it can be proved (cf. [12]) that for Riemannian foliations they are all finite dimensional.…”

Section: Introductionmentioning

confidence: 90%

“…[2][3][4]9,12,14]); in particular, it can be proved (cf. [12]) that for Riemannian foliations they are all finite dimensional. Our examples amount to say that certain compactness conditions in those proofs cannot be dropped, which is by no means obvious (cf.…”

Section: Introductionmentioning

confidence: 90%

See 1 more Smart Citation

Examples of foliations with infinite dimensional special cohomology

Czarnecki

2017

Annali di Matematica

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Cohomology of manifolds with structure group $$U(n) \times O(s)$$

2023

Geom Dedicata

We introduce a new spectral sequence for the study of $${\mathcal {K}}$$ K -manifolds which arises by restricting the spectral sequence of a Riemannian foliation to forms invariant under the flows of $$\{\xi _1,\ldots ,\xi _s\}$$ { ξ 1 , … , ξ s } . We use this sequence to generalize a number of theorems from K-contact geometry to $${\mathcal {K}}$$ K -manifolds. Most importantly we compute the cohomology ring and harmonic forms of $${\mathcal {S}}$$ S -manifolds in terms of primitive basic cohomology and primitive basic harmonic forms (respectively). As an immediate consequence of this we get that the basic cohomology of $${\mathcal {S}}$$ S -manifolds are a topological invariant. We also show that the basic Hodge numbers of $${\mathcal {S}}$$ S -manifolds are invariant under deformations. Finally, we provide similar results for $${\mathcal {C}}$$ C -manifolds.