We review the properties of the Morse-Novikov cohomology and compute it for all known compact complex surfaces with locally conformally Kähler metrics. We present explicit computations for the Inoue surfaces S 0 , S + , S − and classify the locally conformally Kähler (and the tamed locally conformally symplectic) forms on S 0 . We prove the nonexistence of LCK metrics with potential and more generally, of d θ -exact LCK metrics on Inoue surfaces and Oeljeklaus-Toma manifolds.
We study the interplay between the following types of special non-Kähler Hermitian metrics on compact complex manifolds: locally conformally Kähler, k-Gauduchon, balanced and locally conformally balanced and prove that a locally conformally Kähler compact nilmanifold carrying a balanced or a k-Gauduchon metric is necessarily a torus. Combined with the main result in [FV16], this leads to the fact that a compact complex 2-step nilmanifold endowed with whichever two of the following types of metrics: balanced, pluriclosed and locally conformally Kähler is a torus. Moreover, we construct a family of compact nilmanifolds in any dimension carrying both balanced and locally conformally balanced metrics and finally we show a compact complex nilmanifold does not support a left-invariant locally conformally hyperKähler structure.
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...
Abstract. We study the Morse-Novikov cohomology and its almostsymplectic counterpart on manifolds admitting locally conformally symplectic structures. More precisely, we introduce lcs cohomologies and we study elliptic Hodge theory, dualities, Hard Lefschetz Condition. We consider solvmanifolds and Oeljeklaus-Toma manifolds. In particular, we prove that Oeljeklaus-Toma manifolds with precisely one complex place, and under an additional arithmetic condition, satisfy the Mostow property. This holds in particular for the Inoue surface of type S 0 .
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