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

TL;DR

This paper investigates the cohomological properties of LCK manifolds with potential, proving a version of the $dd^c$-lemma with coefficients in powers of the conformal weight bundle, and demonstrating vanishing and degeneration results.

Contribution

It establishes the $dd^c$-lemma with coefficients in powers of the conformal weight bundle for LCK manifolds with potential, including Vaisman manifolds, and proves vanishing of certain cohomologies and spectral sequence degeneration.

Findings

$dd^c$-lemma holds with coefficients in powers of $L$ on LCK manifolds with potential

Vanishing of Dolbeault and Bott-Chern cohomology with coefficients in $L^a$

Degeneration of the Dolbeault-Frolicher spectral sequence with coefficients in any power of $L$

Abstract

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 $L$.