De Rham and Twisted Cohomology of Oeljeklaus–Toma manifolds
Annales de l'Institut Fourier, to appear, 30 p.

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.

Received : 2018-02-13
Revised : 2018-06-12
Accepted : 2018-09-25
Classification:  53C55,  58A12,  55R20,  11R27
Keywords: OT manifold, de Rham cohomology, twisted cohomology, spectral sequence, number field, LCK metric
@unpublished{AIF_0__0_0_A14_0,
     author = {Istrati, Nicolina and Otiman, Alexandra},
     title = {De Rham and Twisted Cohomology of Oeljeklaus--Toma manifolds},
     note = {to appear in \emph{Annales de l'Institut Fourier}},
}
De Rham and Twisted Cohomology of Oeljeklaus–Toma manifolds. Annales de l'Institut Fourier, to appear, 30 p.

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

[2] Apostolov, Vestislav; Dloussky, Georges Locally conformally symplectic structures on compact non-Kähler complex surfaces, Int. Math. Res. Not., Tome 2016 (2016) no. 9, pp. 2717-2747 | Zbl 1404.32032

[3] Bott, Raoul; Tu, Loring W. Differential forms in algebraic topology, Springer, Graduate Texts in Mathematics, Tome 82 (2013) | Zbl 0496.55001

[4] Braunling, Olivier Oeljeklaus–Toma manifolds and arithmetic invariants, Math. Z., Tome 286 (2017) no. 1-2, pp. 291-323 | Zbl 1377.53091

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

[6] Dimca, Alexandru Sheaves in topology, Springer, Universitext (2004) | Zbl 1043.14003

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

[8] Farber, Michael Topology of closed one-forms, American Mathematical Society, Mathematical Surveys and Monographs, Tome 108 (2004) | Zbl 1052.58016

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

[10] Griffiths, Phillip; Harris, Joseph Principles of algebraic geometry, John Wiley & Sons, Pure and Applied Mathematics (1978) | Zbl 0408.14001

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

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

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

[14] Novikov, Sergeĭ P. Multi-valued functions and functionals. An analogue of Morse theory, Sov. Math., Dokl., Tome 24 (1981), pp. 222-226 | Zbl 0505.58011

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

[16] Oeljeklaus, Karl; Toma, Matei Non-Kähler compact complex manifolds associated to number fields, Ann. Inst. Fourier, Tome 55 (2005) no. 1, pp. 161-171 | Zbl 1071.32017

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

[18] Ornea, Liviu; Verbitsky, Misha Oeljeklaus–Toma manifolds admitting no complex subvarieties, Math. Res. Lett., Tome 18 (2011) no. 4, pp. 747-754 | Zbl 1272.53060

[19] Otiman, Alexandra Morse–Novikov cohomology of locally conformally Kähler surfaces, Math. Z., Tome 289 (2018) no. 1-2, pp. 605-628 | Zbl 1397.53086

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

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

[22] Vuletescu, Victor LCK metrics on Oeljeklaus–Toma manifolds versus Kronecker’s theorem, Bull. Math. Soc. Sci. Math. Roum., Tome 57 (2014) no. 2, pp. 225-231 | Zbl 1389.11137