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

抄録

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.