De Rham and Twisted Cohomology of Oeljeklaus–Toma manifolds

Istrati, Nicolina; Otiman, Alexandra

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.

Classification : 53C55, 58A12, 55R20, 11R27
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

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