Morse-Novikov cohomology of locally conformally K\"ahler manifolds

TL;DR

This paper introduces new cohomology invariants for locally conformally Kähler (LCK) manifolds, explores their properties on Vaisman manifolds, and establishes conditions under which LCK manifolds embed into Hopf manifolds.

Contribution

It defines three cohomology invariants for LCK structures and proves their vanishing on Vaisman manifolds, also establishing an embedding theorem for LCK manifolds with vanishing Bott-Chern class.

Findings

Morse-Novikov and Bott-Chern classes vanish on Vaisman manifolds

Vanishing Bott-Chern class implies holomorphic embedding into Hopf manifolds for certain dimensions

LCK structures with coinciding classes differ by a smooth potential

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.