De Rham and Twisted Cohomology of Oeljeklaus–Toma manifolds

Abstract

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. 1 2 NICOLINA ISTRATI AND ALEXANDRA OTIMAN which are not of type ps, 1q. Some progress in this direction has been made by V. Vuletescu [Vul14] and A. Dubickas [Dub14], but the question remains open in general.Concerning the cohomology of OT manifolds, their first Betti number and the second one for a certain subclass of manifolds, called of simple type, were computed in [OT05]. More recently, H. Kasuya computed in [Kas13a] the de Rham cohomology of OT manifolds of type ps, 1q, using their solvmanifold structure. In this note, we use different methods to compute the de Rham cohomology algebra as well as the twisted cohomology of any OT manifold XpK, U q. This is done in terms of numerical invariants coming from U Ă K, and the exact statements are Theorem 3.1 and Theorem 5.1. Theorem 3.1 is proved by two different approaches, one by reducing to the invariant cohomology with respect to a certain compact Lie group, in Section 3, and the other one using the Leray-Serre spectral sequence, in Section 4. This last approach is also used to prove Theorem 5.1 in Section 5. The last section is devoted to a few applications, focusing on the OT manifolds which admit an LCK metric. We compute the explicit cohomology for OT manifolds admitting LCK metrics (Proposition 6.4, Proposition 6.7) and for OT manifolds associated to a certain family of polynomials (Example 6.3). Also, we show that the set of possible Lee classes for an LCK metric on XpK, U q consists of only one element (Proposition 6.5). The problem of characterizing the set of Lee classes was first considered by Tsukada in [Tsu97], who gave a description for Lee classes of LCK metrics on Vaisman manifolds and then by Apostolov and Dloussky in [AD16], where they characterize this set for Hopf surfaces, Inoue surfaces S˘and Kato surfaces. In [Oti18], it is described also for Inoue surfaces of type S 0 and finally, OT manifolds complete the list of known LCK manifolds for which this set is known. Additionally, we determine all the possible twisted classes of LCK forms on OT manifolds (Corollary 6.10), generalizing a result of [Oti18] showing that this class cannot vanish. They all turn out to induce a non-deg...

Locally conformally symplectic convexity

“…Therefore, if H 1 (M, d θ ) = 0, then all special lcs vector fields on M are twisted Hamiltonian. However, the computation of the Morse-Novikov cohomology is not straightforward [19], so this cohomological criterion has limited applicability.…”

Locally conformally Kähler solvmanifolds: a survey

“…It is shown that, after an initial conformal change, the flow converges, in the Gromov-Hausdorff sense, to a torus with a flat Riemannian metric determined by the OT-manifolds themselves. • In [53] both the de Rham cohomology and the Morse-Novikov cohomology of OT manifolds of general type are studied. In fact, the cohomology groups H * dR (X(K, U)) and H * θ (X(K, U)) (for any closed 1-form θ on X(K, U)) are computed in terms of invariants associated to the background number field K. As mentioned previously, OT manifolds with t = 1 admit LCK metrics.…”

“…Recently, Istrati and Otiman determined in [53] the Betti numbers of OT manifolds carrying LCK metrics. Indeed, they show that the Betti numbers b j = dim C H j dR (X, C) of an OT manifold X of type (s, t) admitting some LCK metric are given by…”

“…Regarding the Morse-Novikov cohomology of OT manifolds carrying LCK metrics, it is also shown in [53] that if X = X(K, U) is an OT manifold of type (s, t) then there exists at most one Lee class of a LCK metric. Furthermore, if θ is the Lee form of a LCK structure on X, then the corresponding twisted Betti numbers b θ j = dim C H j θ (X, C) are given by:…”

“…For the Morse-Novikov cohomology, it was shown in [12] that the isomorphism (6) holds for a special class of OT manifolds of type (s, 1), namely, those satisfying the Mostow condition. However, using the description of the Morse-Novikov cohomology of OT manifolds given in [53], Istrati and Otiman show that (6) holds for any OT manifold of type (s, 1), even if the Mostow condition does not hold.…”