Weighted Bott–Chern and Dolbeault cohomology for LCK-manifolds with potential

Abstract

A locally conformally Kähler (LCK) manifold is a complex manifold, with a Kähler structure on its universal covering $\tilde M$, with the deck transform group acting on $\tilde M$ by holomorphic homotheties. One could think of an LCK manifold as of a complex manifold with a Kähler form taking values in a local system $L$, called the conformal weight bundle. The $L$-valued cohomology of $M$ is called Morse–Novikov cohomology; it was conjectured that (just as it happens for Kähler manifolds) the Morse–Novikov complex satisfies the $dd^c$-lemma, which (if true) would have far-reaching consequences for the geometry of LCK manifolds. In particular, this version of $dd^c$-lemma would imply existence of LCK potential on any LCK manifold with vanishing Morse–Novikov class of its $L$-valued Hermitian symplectic form. The $dd^c$-conjecture was disproved for Vaisman manifolds by Goto. We prove that the $dd^c$-lemma is true with coefficients in a sufficiently general power of $L$ on any Vaisman manifold or LCK manifold with potential.