Morse–Novikov cohomology of locally conformally Kähler manifolds
By L. Ornea, Misha Verbitsky. 2007
56 citations recorded by the source corpus.
Abstract A locally conformally Kahler (LCK) manifold is a complex manifold admitting a Kahler covering, with the monodromy acting on this covering by holomorphic homotheties. We define three cohomology invariants, the Lee class, the Morse–Novikov class, and the Bott–Chern class, of an LCK-structure. These invariants play together the same role as the Kahler class in Kahler geometry. If these classes coincide for two LCK-structures, the difference between these structures can be expressed by a smooth potential, similar to the Kahler case. We show that the Morse–Novikov class and the Bott–Chern class of a Vaisman manifold vanish. Moreover, for any LCK-structure on a manifold, admitting a Vaisman structure, we prove that its Morse–Novikov class vanishes. We show that a compact LCK-manifold M with vanishing Bott–Chern class admits a holomorphic embedding into a Hopf manifold, if dim C M ⩾ 3 , a result which parallels the Kodaira embedding theorem.
Subfields: Applied Mathematics, Geometry and Topology
View Paper: https://www.semanticscholar.org/paper/d807d1f91fc936357e0c8bab0d9ab00b8df4fb73
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In Applied Mathematics
In Geometry and Topology
Citations in the corpus
- Vaisman metrics on solvmanifolds and Oeljeklaus–Toma manifolds (2012, background): “In addition to this theorem, if (M,J) admits another LCK (not necessarily Vaisman) form ω0 with Lee form θ0, then θ0 is cohomologous to θ and [ω0]θ0 = 0 (see [24]).”