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Morse-Novikov cohomology of locally conformally Kähler manifolds. (English) Zbl 1161.57015

J. Geom. Phys. 59, No. 3, 295-305 (2009); erratum ibid. 107, 92-98 (2016).
Three cohomology invariants of a locally conformal Kähler (LCK) structure are defined. These are the Lee class, the Morse-Novikov class and the Bott-Chern class which play together the same role as the Kähler class in Kähler geometry. It is proved that if these classes coincide for two LCK–structures then the difference between these structures can be expressed by a smooth potential. Also, it is shown that the Morse-Novikov class and the Bott-Chern class of a Vaisman manifold vanish. In fact the Morse-Novikov class of any LCK–structure on a manifold admitting a Vaisman structure vanishes. Finally, it is proved that a compact LCK–manifold with complex dimension at least 3 and with vanishing Bott-Chern class admits a holomorphic embedding into a Hopf manifold.
For an erratum to this paper see [Zbl 1347.53058].

MSC:

57R20 Characteristic classes and numbers in differential topology
53C56 Other complex differential geometry

Citations:

Zbl 1347.53058
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References:

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