• A
  • A
  • A
  • ABC
  • ABC
  • ABC
  • А
  • А
  • А
  • А
  • А
Regular version of the site

Article

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

Journal of the Mathematical Society of Japan. 2018. Vol. 70. No. 1. P. 409-422.
Verbitsky M., Liviu O., Vuletescu V.

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.