De Rham and twisted cohomology of Oeljeklaus-Toma manifolds. (Cohomologie de De Rham et twistée des variétés d’Oeljeklaus-Toma.) (English. French summary) Zbl 1426.53083
Summary: 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.
MSC:
| 53C56 | Other complex differential geometry |
| 53C55 | Global differential geometry of Hermitian and Kählerian manifolds |
| 58A12 | de Rham theory in global analysis |
| 55R20 | Spectral sequences and homology of fiber spaces in algebraic topology |
| 11R27 | Units and factorization |
Keywords:
OT manifold; De Rham cohomology; twisted cohomology; spectral sequence; number field; LCK metricReferences:
| [1] | Angella, Daniele; Otiman, Alexandra; Tardini, Nicoletta, Cohomologies of locally conformally symplectic manifolds and solvmanifolds, Ann. Global Anal. Geom., 53, 1, 67-96 (2018) · Zbl 1394.32022 · doi:10.1007/s10455-017-9568-y |
| [2] | Apostolov, Vestislav; Dloussky, Georges, Locally conformally symplectic structures on compact non-Kähler complex surfaces, Int. Math. Res. Not., 2016, 9, 2717-2747 (2016) · Zbl 1404.32032 · doi:10.1093/imrn/rnv211 |
| [3] | Bott, Raoul; Tu, Loring W., Differential forms in algebraic topology, 82 (2013), Springer · Zbl 0496.55001 |
| [4] | Braunling, Olivier, Oeljeklaus-Toma manifolds and arithmetic invariants, Math. Z., 286, 1-2, 291-323 (2017) · Zbl 1377.53091 · doi:10.1007/s00209-016-1763-1 |
| [5] | De León, Manuel; López, Belén; Marrero, Juan C.; Padrón, Edith, On the computation of the Lichnerowicz-Jacobi cohomology, J. Geom. Phys., 44, 4, 507-522 (2003) · Zbl 1092.53060 · doi:10.1016/S0393-0440(02)00056-6 |
| [6] | Dimca, Alexandru, Sheaves in topology (2004), Springer · Zbl 1043.14003 |
| [7] | Dubickas, Arturas, Nonreciprocal units in a number field with an application to Oeljeklaus-Toma manifolds, New York J. Math., 20, 257-274 (2014) · Zbl 1290.11148 |
| [8] | Farber, Michael, Topology of closed one-forms, 108 (2004), American Mathematical Society · Zbl 1052.58016 |
| [9] | Goto, Ryushi, On the stability of locally conformal Kähler structures, J. Math. Soc. Japan, 66, 4, 1375-1401 (2014) · Zbl 1310.53062 · doi:10.2969/jmsj/06641375 |
| [10] | Griffiths, Phillip; Harris, Joseph, Principles of algebraic geometry (1978), John Wiley & Sons · Zbl 0408.14001 |
| [11] | Inoue, Masahisa, On surfaces of class VII 0, Invent. Math., 24, 4, 269-310 (1974) · Zbl 0283.32019 · doi:10.1007/BF01425563 |
| [12] | Kasuya, Hisashi, Minimal models, formality, and hard Lefschetz properties of solvmanifolds with local systems, J. Differ. Geom., 93, 2, 269-297 (2013) · Zbl 1373.53069 |
| [13] | Kasuya, Hisashi, Vaisman metrics on solvmanifolds and Oeljeklaus-Toma manifolds, Bull. Lond. Math. Soc., 45, 1, 15-26 (2013) · Zbl 1262.53061 · doi:10.1112/blms/bds057 |
| [14] | Novikov, Sergeĭ P., Multi-valued functions and functionals. An analogue of Morse theory, Sov. Math., Dokl., 24, 222-226 (1981) · Zbl 0505.58011 |
| [15] | Novikov, Sergeĭ P., The Hamiltonian formalism and a multi-valued analogue of Morse theory, Russ. Math. Surv., 37, 1-56 (1982) · Zbl 0571.58011 · doi:10.1070/RM1982v037n05ABEH004020 |
| [16] | Oeljeklaus, Karl; Toma, Matei, Non-Kähler compact complex manifolds associated to number fields, Ann. Inst. Fourier, 55, 1, 161-171 (2005) · Zbl 1071.32017 · doi:10.5802/aif.2093 |
| [17] | Ornea, Liviu; Verbitsky, Misha, Locally conformal Kähler manifolds with potential, Math. Ann., 348, 1, 25-33 (2010) · Zbl 1213.53090 |
| [18] | Ornea, Liviu; Verbitsky, Misha, Oeljeklaus-Toma manifolds admitting no complex subvarieties, Math. Res. Lett., 18, 4, 747-754 (2011) · Zbl 1272.53060 · doi:10.4310/MRL.2011.v18.n4.a12 |
| [19] | Otiman, Alexandra, Morse-Novikov cohomology of locally conformally Kähler surfaces, Math. Z., 289, 1-2, 605-628 (2018) · Zbl 1397.53086 · doi:10.1007/s00209-017-1968-y |
| [20] | Parton, Maurizio; Vuletescu, Victor, Examples of non-trivial rank in locally conformal Kähler geometry, Math. Z., 270, 1-2, 179-187 (2012) · Zbl 1242.32011 · doi:10.1007/s00209-010-0791-5 |
| [21] | Tsukada, Kazumi, Holomorphic maps of compact generalized Hopf manifolds, Geom. Dedicata, 68, 1, 61-71 (1997) · Zbl 0916.53035 · doi:10.1023/A:1004949925097 |
| [22] | Vuletescu, Victor, LCK metrics on Oeljeklaus-Toma manifolds versus Kronecker’s theorem, Bull. Math. Soc. Sci. Math. Roum., 57, 2, 225-231 (2014) · Zbl 1389.11137 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
