Morse-Novikov cohomology of locally conformally Kähler surfaces

Otiman, Alexandra

doi:10.1007/s00209-017-1968-y

Cited by 23 publications

(19 citation statements)

References 31 publications

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“…Finally, since in Theorem 5.1 we represented the corresponding cohomologies by invariant forms with respect to the action of G described in [Kas13b], we obtain as a consequence that for all OT manifolds X of type ps, tq, we have the isomorphism H k θ pXq -H k θ pgq, although they might not all satisfy the Mostow condition. In [Oti18] it was proven that there are no d θ -exact metrics on OT manifolds of type ps, 1q. We give next a generalization of this result, in which we determine all the possible LCK classes in H 2 θ .…”

Section: Applications and Examplesmentioning

confidence: 99%

De Rham and Twisted Cohomology of Oeljeklaus–Toma manifolds

Istrati¹,

Otiman²

2019

Annales De l'Institut Fourier

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Oeljeklaus-Toma (OT) manifolds are complex non-Kähler manifolds whose construction arises from specific number fields. In this note, we compute their de Rham cohomology in terms of invariants associated to the background number field. This is done by two distinct approaches, one by averaging over a certain compact group, and the other one using the Leray-Serre spectral sequence. In addition, we compute also their twisted cohomology. As an application, we show that the low degree Chern classes of any complex vector bundle on an OT manifold vanish in the real cohomology. Other applications concern the OT manifolds admitting locally conformally Kähler (LCK) metrics: we show that there is only one possible Lee class of an LCK metric, and we determine all the possible twisted classes of an LCK metric, which implies the nondegeneracy of certain Lefschetz maps in cohomology. 1 2 NICOLINA ISTRATI AND ALEXANDRA OTIMAN which are not of type ps, 1q. Some progress in this direction has been made by V. Vuletescu [Vul14] and A. Dubickas [Dub14], but the question remains open in general.Concerning the cohomology of OT manifolds, their first Betti number and the second one for a certain subclass of manifolds, called of simple type, were computed in [OT05]. More recently, H. Kasuya computed in [Kas13a] the de Rham cohomology of OT manifolds of type ps, 1q, using their solvmanifold structure. In this note, we use different methods to compute the de Rham cohomology algebra as well as the twisted cohomology of any OT manifold XpK, U q. This is done in terms of numerical invariants coming from U Ă K, and the exact statements are Theorem 3.1 and Theorem 5.1. Theorem 3.1 is proved by two different approaches, one by reducing to the invariant cohomology with respect to a certain compact Lie group, in Section 3, and the other one using the Leray-Serre spectral sequence, in Section 4. This last approach is also used to prove Theorem 5.1 in Section 5. The last section is devoted to a few applications, focusing on the OT manifolds which admit an LCK metric. We compute the explicit cohomology for OT manifolds admitting LCK metrics (Proposition 6.4, Proposition 6.7) and for OT manifolds associated to a certain family of polynomials (Example 6.3). Also, we show that the set of possible Lee classes for an LCK metric on XpK, U q consists of only one element (Proposition 6.5). The problem of characterizing the set of Lee classes was first considered by Tsukada in [Tsu97], who gave a description for Lee classes of LCK metrics on Vaisman manifolds and then by Apostolov and Dloussky in [AD16], where they characterize this set for Hopf surfaces, Inoue surfaces S˘and Kato surfaces. In [Oti18], it is described also for Inoue surfaces of type S 0 and finally, OT manifolds complete the list of known LCK manifolds for which this set is known. Additionally, we determine all the possible twisted classes of LCK forms on OT manifolds (Corollary 6.10), generalizing a result of [Oti18] showing that this class cannot vanish. They all turn out to induce a non-deg...

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Section: Applications and Examplesmentioning

confidence: 99%

De Rham and Twisted Cohomology of Oeljeklaus–Toma manifolds

Istrati¹,

Otiman²

2019

Annales De l'Institut Fourier

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“…The authors consider locally conformally symplectic structures (ω, ϑ ) on compact complex surfaces (M, J) such that ω tames J, i. e. the (1, 1)-part of ω is positive definite. The Morse-Novikov cohomology of locally conformally Kähler surfaces has been investigated in [108]. Results on the deformations of Lee classes of locally conformally Kähler structures have been obtained in [51].…”

Section: Kähler and Locally Conformally Kähler Geometrymentioning

confidence: 99%

Locally conformally symplectic and Kähler geometry

Bazzoni

2018

EMS Surv. Math. Sci.

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The goal of this note is to give an introduction to locally conformally symplectic and Kähler geometry. In particular, Sections 1 and 3 aim to provide the reader with enough mathematical background to appreciate this kind of geometry. The reference book for locally conformally Kähler geometry is [36] by Sorin Dragomir and Liviu Ornea. Many progresses in this field, however, were accomplished after the publication of this book, hence are not contained there -see the introduction of [97]. On the other hand, there is no book on locally conformally symplectic geometry and many recent advances lie scattered in the literature. Sections 2 and 4 would like to demonstrate how these geometries can be used to give precise mathematical formulations to ideas deeply rooted in classical and modern Physics.

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“…See e.g. the explicit computations on Inoue surfaces in [Oti16], where the Morse-Novikov cohomology allows to distinguish between Inoue surfaces of type S + and S − , even if they have the same Betti numbers. So, it may be useful to understand the cohomology H • ϑ (X) varying [ϑ] ∈ H 1 (X; R); in particular one can study, for example, H • k·ϑ (X) varying k ∈ R for a fixed [ϑ] ∈ H 1 (X; R).…”

Section: Introductionmentioning

confidence: 99%

“…Following the same pattern as [Bry88,Mat95,Yan96,Mer98,Gui01,Cav05,TY12a], we study ellipticHodge-theory, and we get some results concerning Poincaré dualities, see Proposition 2.3 and Theorem 2.4, and Hard Lefschetz Condition, see Theorem 2.6 and Theorem 2.7. Finally, we study some explicit examples, on nilmanifolds (Kodaira-Thurston surface [Kod64,Thu76]) and solvmanifolds (Inoue surfaces of type S + [Ino74], for which see also [Oti16], and OeljeklausToma manifolds [OT05]). …”

Section: Introductionmentioning

confidence: 99%

“…The case of Inoue surfaces is interesting because two subclasses, S ± , are completely-solvable, falling thus under the scope of the Hattori theorem; however this is not the case of the subclass S 0 . In [Oti16], the computations of the cohomology are done without using the structure of solvmanifold, but instead with a "twisted" version of the Mayer-Vietoris sequence. We prove here that Inoue surfaces of type S 0 and, more in general, certain Oeljeklaus-Toma manifolds with precisely one complex place satisfy the Mostow condition, see Proposition 4.2 and Theorem 4.3 respectively.…”

Section: Introductionmentioning

confidence: 99%

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Cohomologies of locally conformally symplectic manifolds and solvmanifolds

Angella¹,

Otiman²,

Tardini³

2017

Ann Glob Anal Geom

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Abstract. We study the Morse-Novikov cohomology and its almostsymplectic counterpart on manifolds admitting locally conformally symplectic structures. More precisely, we introduce lcs cohomologies and we study elliptic Hodge theory, dualities, Hard Lefschetz Condition. We consider solvmanifolds and Oeljeklaus-Toma manifolds. In particular, we prove that Oeljeklaus-Toma manifolds with precisely one complex place, and under an additional arithmetic condition, satisfy the Mostow property. This holds in particular for the Inoue surface of type S 0 .

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Morse-Novikov cohomology of locally conformally Kähler surfaces

Cited by 23 publications

References 31 publications

De Rham and Twisted Cohomology of Oeljeklaus–Toma manifolds

De Rham and Twisted Cohomology of Oeljeklaus–Toma manifolds

Locally conformally symplectic and Kähler geometry

Cohomologies of locally conformally symplectic manifolds and solvmanifolds

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