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.
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.
Revised:
Accepted:
Published online:
DOI: 10.5802/aif.3288
Keywords: OT manifold, de Rham cohomology, twisted cohomology, spectral sequence, number field, LCK metric
Mot clés : variété OT, cohomologie de De Rham, cohomologie twistée, suite spectrale, corps de nombres, métrique LCK
@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 = {http://archive.numdam.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 - http://archive.numdam.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 http://archive.numdam.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, Volume 69 (2019) no. 5, pp. 2037-2066. doi : 10.5802/aif.3288. http://archive.numdam.org/articles/10.5802/aif.3288/
[1] Cohomologies of locally conformally symplectic manifolds and solvmanifolds, Ann. Global Anal. Geom., Volume 53 (2018) no. 1, pp. 67-96 | DOI | MR | Zbl
[2] Locally conformally symplectic structures on compact non-Kähler complex surfaces, Int. Math. Res. Not., Volume 2016 (2016) no. 9, pp. 2717-2747 | DOI | Zbl
[3] Differential forms in algebraic topology, Graduate Texts in Mathematics, 82, Springer, 2013 | Zbl
[4] Oeljeklaus–Toma manifolds and arithmetic invariants, Math. Z., Volume 286 (2017) no. 1-2, pp. 291-323 | DOI | MR | Zbl
[5] On the computation of the Lichnerowicz–Jacobi cohomology, J. Geom. Phys., Volume 44 (2003) no. 4, pp. 507-522 | DOI | MR | Zbl
[6] Sheaves in topology, Universitext, Springer, 2004 | Zbl
[7] 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] Topology of closed one-forms, Mathematical Surveys and Monographs, 108, American Mathematical Society, 2004 | MR | Zbl
[9] On the stability of locally conformal Kähler structures, J. Math. Soc. Japan, Volume 66 (2014) no. 4, pp. 1375-1401 | DOI | MR | Zbl
[10] Principles of algebraic geometry, Pure and Applied Mathematics, John Wiley & Sons, 1978 | Zbl
[11] On surfaces of class VII 0, Invent. Math., Volume 24 (1974) no. 4, pp. 269-310 | DOI | MR | Zbl
[12] 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] Vaisman metrics on solvmanifolds and Oeljeklaus–Toma manifolds, Bull. Lond. Math. Soc., Volume 45 (2013) no. 1, pp. 15-26 | DOI | MR | Zbl
[14] Multi-valued functions and functionals. An analogue of Morse theory, Sov. Math., Dokl., Volume 24 (1981), pp. 222-226 | Zbl
[15] The Hamiltonian formalism and a multi-valued analogue of Morse theory, Russ. Math. Surv., Volume 37 (1982), pp. 1-56 | DOI
[16] Non-Kähler compact complex manifolds associated to number fields, Ann. Inst. Fourier, Volume 55 (2005) no. 1, pp. 161-171 | DOI | Zbl
[17] Locally conformal Kähler manifolds with potential, Math. Ann., Volume 348 (2010) no. 1, pp. 25-33 | Zbl
[18] Oeljeklaus–Toma manifolds admitting no complex subvarieties, Math. Res. Lett., Volume 18 (2011) no. 4, pp. 747-754 | DOI | MR | Zbl
[19] Morse–Novikov cohomology of locally conformally Kähler surfaces, Math. Z., Volume 289 (2018) no. 1-2, pp. 605-628 | DOI | MR | Zbl
[20] Examples of non-trivial rank in locally conformal Kähler geometry, Math. Z., Volume 270 (2012) no. 1-2, pp. 179-187 | DOI | Zbl
[21] Holomorphic maps of compact generalized Hopf manifolds, Geom. Dedicata, Volume 68 (1997) no. 1, pp. 61-71 | DOI | MR | Zbl
[22] 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
Cited by Sources: