Morse-Novikov cohomology of locally conformally Kähler manifolds

Abstract

A locally conformally Kahler (LCK) manifold is a complex manifold admitting a Kahler covering, with the monodromy acting on this covering by homotheties. We define three cohomology invariants, the Lee class, the Morse-Novikov class, and the Bott-Chern class, of an LCK-structure. These invariants together play the same role as the Kahler class in Kahler geometry. If these classes for two LCK-structures coincide, 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 vanishes. Moreover, for any LCK-structure on a Vaisman manifold, 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 to a Hopf manifold, if $\dim_\C M \geq 3$, a result which parallels the Kodaira embedding theorem.

22 pages. Version 4.0, minor corrections, clarifications and typos. To appear in Journal of Geometry and Physics

References in corpus (3)

• Locally conformally Kähler manifolds. A selection of results
• Vanishing theorems for locally conformal hyperkaehler manifolds
• Immersion theorem for Vaisman manifolds

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