De Rham and Twisted Cohomology of Oeljeklaus–Toma manifolds

Abstract

Oeljeklaus-Toma (OT) manifolds are complex non-Kahler 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 using invariant cohomology and the other one using the Leray-Serre spectral sequence. In addition, we compute also their Morse-Novikov 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 Kahler (LCK) metrics: we show that there is only one possible Lee class of an LCK metric, and we determine all the possible Morse-Novikov classes of an LCK metric, which implies the nondegeneracy of certain Lefschetz maps in cohomology.