De Rham and twisted cohomology of Oeljeklaus-Toma manifolds

TL;DR

This paper computes the de Rham and Morse-Novikov cohomology of Oeljeklaus-Toma manifolds, revealing their topological invariants and implications for complex vector bundles and locally conformally Kähler metrics.

Contribution

It provides explicit cohomology computations for OT manifolds using two methods and explores their geometric and topological properties, including LCK metrics and Chern classes.

Findings

De Rham cohomology expressed via number field invariants.

All low degree Chern classes of vector bundles vanish in real cohomology.

Uniqueness of the Lee class for LCK metrics on OT manifolds.

Abstract

Oeljeklaus-Toma (OT) manifolds are complex non-K\"ahler 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 K\"ahler (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.