no. 5
De Rham and Twisted Cohomology of Oeljeklaus–Toma manifolds
[Cohomologie de De Rham et twistée des variétés d’Oeljeklaus–Toma]
Istrati, Nicolina1 ; Otiman, Alexandra2
1 Univ. Paris Diderot, Sorbonne Paris Cité Institut de Mathématiques de Jussieu-Paris Rive Gauche Case 7012, 75205 Paris Cedex 13 (France)
2 Institute of Mathematics “Simion Stoilow” of the Romanian Academy 21, Calea Grivitei, 010702, Bucharest (Romania) and University of Bucharest, Research Center in Geometry, Topology and Algebra Faculty of Mathematics and Computer Science 14 Academiei Str., Bucharest (Romania) and Max Planck Institut für Mathematik Vivatsgasse 7, 53111 Bonn (Germany)
Annales de l'Institut Fourier, Tome 69 (2019) no. 5, pp. 2037-2066.

Les variétés d’Oeljeklaus–Toma (OT) sont des variétés complexes non-kähleriennes qui sont construites à partir des corps de nombres. Dans cet article, nous calculons leur cohomologie de De Rham en termes d’invariants associés au corps de nombres associés. Nous faisons cela de deux manières différentes, l’une en moyennant sur un certain groupe compact, et l’autre en utilisant la suite spectrale de Leray–Serre. De plus, nous déterminons aussi leur cohomologie twistée. Comme application, nous montrons que les classes de Chern de bas degré de tout fibré vectoriel complexe sur une variété OT s’annulent dans la cohomologie réelle. D’autres applications concernent les variétés OT admettant des métriques localement conformément kähleriennes (LCK) : nous montrons qu’il existe une unique classe de Lee possible pour une métrique LCK et nous determinons toutes les classes twistées des métriques LCK, ce qui implique que certains morphismes de Lefschetz en cohomologie sont non-dégénérés.

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.

Reçu le :
Révisé le :
Accepté le :
Publié le :
DOI : 10.5802/aif.3288
Classification : 53C55, 58A12, 55R20, 11R27
Keywords: OT manifold, de Rham cohomology, twisted cohomology, spectral sequence, number field, LCK metric
Mots-clés : variété OT, cohomologie de De Rham, cohomologie twistée, suite spectrale, corps de nombres, métrique LCK

Istrati, Nicolina 1 ; Otiman, Alexandra 2

1 Univ. Paris Diderot, Sorbonne Paris Cité Institut de Mathématiques de Jussieu-Paris Rive Gauche Case 7012, 75205 Paris Cedex 13 (France)
2 Institute of Mathematics “Simion Stoilow” of the Romanian Academy 21, Calea Grivitei, 010702, Bucharest (Romania) and University of Bucharest, Research Center in Geometry, Topology and Algebra Faculty of Mathematics and Computer Science 14 Academiei Str., Bucharest (Romania) and Max Planck Institut für Mathematik Vivatsgasse 7, 53111 Bonn (Germany)
Licence : CC-BY-ND 4.0
Droits d'auteur : Les auteurs conservent leurs droits 
@article{AIF_2019__69_5_2037_0,
     author = {Istrati, Nicolina and Otiman, Alexandra},
     title = {De {Rham} and {Twisted} {Cohomology} of {Oeljeklaus{\textendash}Toma} manifolds},
     journal = {Annales de l'Institut Fourier},
     pages = {2037--2066},
     publisher = {Association des Annales de l{\textquoteright}institut Fourier},
     volume = {69},
     number = {5},
     year = {2019},
     doi = {10.5802/aif.3288},
     language = {en},
     url = {https://aif.centre-mersenne.org/articles/10.5802/aif.3288/}
}
TY  - JOUR
AU  - Istrati, Nicolina
AU  - Otiman, Alexandra
TI  - De Rham and Twisted Cohomology of Oeljeklaus–Toma manifolds
JO  - Annales de l'Institut Fourier
PY  - 2019
SP  - 2037
EP  - 2066
VL  - 69
IS  - 5
PB  - Association des Annales de l’institut Fourier
UR  - https://aif.centre-mersenne.org/articles/10.5802/aif.3288/
DO  - 10.5802/aif.3288
LA  - en
ID  - AIF_2019__69_5_2037_0
ER  - 
%0 Journal Article
%A Istrati, Nicolina
%A Otiman, Alexandra
%T De Rham and Twisted Cohomology of Oeljeklaus–Toma manifolds
%J Annales de l'Institut Fourier
%D 2019
%P 2037-2066
%V 69
%N 5
%I Association des Annales de l’institut Fourier
%U https://aif.centre-mersenne.org/articles/10.5802/aif.3288/
%R 10.5802/aif.3288
%G en
%F AIF_2019__69_5_2037_0
Istrati, Nicolina; Otiman, Alexandra. De Rham and Twisted Cohomology of Oeljeklaus–Toma manifolds. Annales de l'Institut Fourier, Tome 69 (2019) no. 5, pp. 2037-2066. doi : 10.5802/aif.3288. https://aif.centre-mersenne.org/articles/10.5802/aif.3288/

[1] Angella, Daniele; Otiman, Alexandra; Tardini, Nicoletta Cohomologies of locally conformally symplectic manifolds and solvmanifolds, Ann. Global Anal. Geom., Volume 53 (2018) no. 1, pp. 67-96 | DOI | MR | Zbl

[3] Bott, Raoul; Tu, Loring W. Differential forms in algebraic topology, Graduate Texts in Mathematics, 82, Springer, 2013 | Zbl

[4] Braunling, Olivier Oeljeklaus–Toma manifolds and arithmetic invariants, Math. Z., Volume 286 (2017) no. 1-2, pp. 291-323 | DOI | MR | Zbl

[5] De León, Manuel; López, Belén; Marrero, Juan C.; Padrón, Edith On the computation of the Lichnerowicz–Jacobi cohomology, J. Geom. Phys., Volume 44 (2003) no. 4, pp. 507-522 | DOI | MR | Zbl

[6] Dimca, Alexandru Sheaves in topology, Universitext, Springer, 2004 | Zbl

[7] Dubickas, Arturas Nonreciprocal units in a number field with an application to Oeljeklaus–Toma manifolds, New York J. Math., Volume 20 (2014), pp. 257-274 | MR | Zbl

[8] Farber, Michael Topology of closed one-forms, Mathematical Surveys and Monographs, 108, American Mathematical Society, 2004 | MR | Zbl

[9] Goto, Ryushi On the stability of locally conformal Kähler structures, J. Math. Soc. Japan, Volume 66 (2014) no. 4, pp. 1375-1401 | DOI | MR | Zbl

[10] Griffiths, Phillip; Harris, Joseph Principles of algebraic geometry, Pure and Applied Mathematics, John Wiley & Sons, 1978 | Zbl

[11] Inoue, Masahisa On surfaces of class VII 0, Invent. Math., Volume 24 (1974) no. 4, pp. 269-310 | DOI | MR | Zbl

[12] Kasuya, Hisashi Minimal models, formality, and hard Lefschetz properties of solvmanifolds with local systems, J. Differ. Geom., Volume 93 (2013) no. 2, pp. 269-297 | MR | Zbl

[13] Kasuya, Hisashi Vaisman metrics on solvmanifolds and Oeljeklaus–Toma manifolds, Bull. Lond. Math. Soc., Volume 45 (2013) no. 1, pp. 15-26 | DOI | MR | Zbl

[14] Novikov, Sergeĭ P. Multi-valued functions and functionals. An analogue of Morse theory, Sov. Math., Dokl., Volume 24 (1981), pp. 222-226 | Zbl

[15] Novikov, Sergeĭ P. The Hamiltonian formalism and a multi-valued analogue of Morse theory, Russ. Math. Surv., Volume 37 (1982), pp. 1-56 | DOI

[16] Oeljeklaus, Karl; Toma, Matei Non-Kähler compact complex manifolds associated to number fields, Ann. Inst. Fourier, Volume 55 (2005) no. 1, pp. 161-171 | DOI | Zbl

[17] Ornea, Liviu; Verbitsky, Misha Locally conformal Kähler manifolds with potential, Math. Ann., Volume 348 (2010) no. 1, pp. 25-33 | Zbl

[18] Ornea, Liviu; Verbitsky, Misha Oeljeklaus–Toma manifolds admitting no complex subvarieties, Math. Res. Lett., Volume 18 (2011) no. 4, pp. 747-754 | DOI | MR | Zbl

[19] Otiman, Alexandra Morse–Novikov cohomology of locally conformally Kähler surfaces, Math. Z., Volume 289 (2018) no. 1-2, pp. 605-628 | DOI | MR | Zbl

[20] Parton, Maurizio; Vuletescu, Victor Examples of non-trivial rank in locally conformal Kähler geometry, Math. Z., Volume 270 (2012) no. 1-2, pp. 179-187 | DOI | Zbl

[21] Tsukada, Kazumi Holomorphic maps of compact generalized Hopf manifolds, Geom. Dedicata, Volume 68 (1997) no. 1, pp. 61-71 | DOI | MR | Zbl

[22] Vuletescu, Victor LCK metrics on Oeljeklaus–Toma manifolds versus Kronecker’s theorem, Bull. Math. Soc. Sci. Math. Roum., Volume 57 (2014) no. 2, pp. 225-231 | MR | Zbl

Cité par Sources :