The Frolicher-type inequalities of foliations

JOURNAL OF GEOMETRY AND PHYSICS

卷:114

Article

Razny, Pawel¹ 

[1] Jagiellonian Univ, Fac Math & Comp Sci, Inst Math, Krakow, Poland

关键词: Foliations;    Transverse structures;    Basic cohomology;    Orbifolds;   

DOI  :  10.1016/j.geomphys.2017.01.005

来源: Elsevier

【 摘 要 】

The purpose of this article is to adapt the Frolicher-type inequality, stated and proven for complex and symplectic manifolds in Angella and Tomassini (2015), to the case of transversely holomorphic and symplectic foliations. These inequalities provide a criterion for checking whether a foliation transversely satisfies the partial derivative(partial derivative) over bar -lemma and the dd(Lambda)-lemma (i.e. whether the basic forms of a given foliation satisfy them). These lemmas are linked to such properties as the formality of the basic de Rham complex of a foliation and the transverse hard Lefschetz property. In particular they provide an obstruction to the existence of a transverse Miller structure for a given foliation. In the second section we will provide some information concerning the d'd ''-lemma for a given double complex (K-center dot,K-center dot d', d '') and state the main results from Angella and Tomassini (2015). We will also recall some basic facts and definitions concerning foliations. In the third section we treat the case of transversely holomorphic foliations. We also give a brief review of some properties of the basic Bott-Chern and Aeppli cohomology theories. In Section 4 we prove the symplectic version of the Frolicher-type inequality. The final 3 sections of this paper are devoted to the applications of our main theorems. In them we verify the aforementioned lemmas for some simple examples, give the orbifold versions of the Frolicher-type inequalities and show that transversely Kahler foliations satisfy both the partial derivative(partial derivative) over bar -lemma and the de-lemma (or in other words that our main theorems provide an obstruction to the existence of a transversely Miller structure). (C) 2017 Elsevier B.V. All rights reserved.

【 授权许可】

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