On Bott-Chern cohomology of compact complex surfaces

Angella, Daniele; Dloussky, Georges; Томассини, Адриано

doi:10.1007/s10231-014-0458-7

Cited by 35 publications

(72 citation statements)

References 18 publications

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“…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%

Examples of foliations with infinite dimensional special cohomology

Czarnecki

Raźny

2017

Annali di Matematica

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Section: Introductionmentioning

confidence: 90%

Examples of foliations with infinite dimensional special cohomology

Czarnecki

Raźny

2017

Annali di Matematica

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“…Nevertheless we can be even more precise, indeed, it is proven in [30] that ∆ 1 vanishes on any compact complex surface (see [10] for explicit examples). This is not true in higher dimension.…”

Section: 1mentioning

confidence: 98%

Cohomological Aspects on Complex and Symplectic Manifolds

Tardini

2017

Complex and Symplectic Geometry

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We discuss how quantitative cohomological informations could provide qualitative properties on complex and symplectic manifolds. In particular we focus on the Bott-Chern and the Aeppli cohomology groups in both cases, since they represent useful tools in studying non Kähler geometry. We give an overview on the comparisons among the dimensions of the cohomology groups that can be defined and we show how we reach the ∂∂-lemma in complex geometry and the Hard-Lefschetz condition in symplectic geometry. For more details we refer to [6] and [29].

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“…On the other hand,

H_{normalBC}^{1, 1} false(V false) ≃ C

(cf. ) and the Bott‐Chern class

c_{normalBC}^{1} false(L false)

is a generator. We show that it is localized at one of the fibers

C

ρ

.…”

Section: An Examplementioning

confidence: 99%

“…It is a powerful tool in the study of non-Kähler manifolds (cf. [3,4] and references therin) and is also related to arithmetic characteristic classes (cf. [11,14]).…”

mentioning

confidence: 99%

Localization of Bott‐Chern classes and Hermitian residues

Corrêa

Suwa

2019

J. London Math. Soc.

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We develop a theory of Čech‐Bott‐Chern cohomology and in this context we naturally come up with the relative Bott‐Chern cohomology. In fact, Bott‐Chern cohomology has two relatives and they all arise from a single complex. Thus, we study these three cohomologies in a unified way and obtain a long exact sequence involving the three. We then study the localization problem of characteristic classes in the relative Bott‐Chern cohomology. For this, we define the cup product and integration in our framework and we discuss local and global duality morphisms. After reviewing some materials on connections, we give a vanishing theorem relevant to our localization. With these, we prove a residue theorem for vector bundles admitting a Hermitian connection compatible with an action of the non‐singular part of a singular distribution. As a typical case, we discuss the action of a distribution on the normal bundle of an invariant submanifold (the so‐called Camacho–Sad action) and give a specific example.

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On Bott-Chern cohomology of compact complex surfaces

Abstract: Abstract. We study Bott-Chern cohomology on compact complex non-Kähler surfaces. In particular, we compute such a cohomology for compact complex surfaces in class VII and for compact complex surfaces diffeomorphic to solvmanifolds.

Cited by 35 publications

References 18 publications

Examples of foliations with infinite dimensional special cohomology

Examples of foliations with infinite dimensional special cohomology

Cohomological Aspects on Complex and Symplectic Manifolds

Localization of Bott‐Chern classes and Hermitian residues

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