  A locally conformally Kahler (LCK) manifold is a complex manifold with a
Kahler structure on its covering and the deck transform group acting on it 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 $dd^c$-lemma. If
true, it would have far-reaching consequences for the geometry of LCK
manifolds. Counterexamples to the Morse-Novikov $dd^c$-lemma on Vaisman
manifolds were found by R. Goto. We prove that $dd^c$-lemma is true with
coefficients in a sufficiently general power $L^a$ of $L$ on any LCK manifold
with potential (this includes Vaisman manifolds). We also prove vanishing of
Dolbeault and Bott-Chern cohomology with coefficients in $L^a$. The same
arguments are used to prove degeneration of the Dolbeault-Frohlicher spectral
sequence with coefficients in any power of $L$.
