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

A locally conformally Kahler (LCK) manifold is a complex manifold, with a Kahler structure on its universal covering (M) over tilde, with the deck transform group acting on (M) over tilde by holomorphic homotheties. One could think of an LCK manifold as of a complex manifold with a Kahler 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 Kahler manifolds) the Morse-Novikov complex satisfies the de(c)-lemma, which (if true) would have far-reaching consequences for the geometry of LCK manifolds. In particular, this version of de(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 de(c)-conjecture was disproved for Vaisman manifolds by Goto. We prove that the de(c)-lemma is true with coefficients in a sufficiently general power of L on any Vaisman manifold or LCK manifold with potential.