Morse-Novikov cohomology of locally conformally K\"ahler surfaces

TL;DR

This paper reviews Morse-Novikov cohomology for compact complex surfaces with locally conformally K"ahler metrics, providing explicit calculations for Inoue surfaces and classifying certain forms, while proving nonexistence results for specific metrics.

Contribution

It offers explicit Morse-Novikov cohomology computations for all known LCK surfaces and classifies LCK forms on Inoue surfaces, advancing understanding of their geometric structures.

Findings

Computed Morse-Novikov cohomology for all known LCK surfaces.

Classified LCK and tamed LCK forms on Inoue surfaces.

Proved nonexistence of certain LCK metrics with potential.

Abstract

We review the properties of the Morse-Novikov cohomology and compute it for all known compact complex surfaces with locally conformally K\"ahler metrics. We present explicit computations for the Inoue surfaces $\mathcal{S}^0$, $\mathcal{S}^+$, $\mathcal{S}^-$ and classify the locally conformally K\"ahler (and the tamed locally conformally symplectic) forms on $\mathcal{S}^0$. We prove the nonexistence of LCK metrics with potential and more generally, of $d_\theta$-exact LCK metrics on Inoue surfaces and Oeljeklaus-Toma manifolds.