Cohomology theories on locally conformal symplectic manifolds

TL;DR

This paper introduces primitive cohomology groups for locally conformal symplectic manifolds, explores their relation to Lichnerowicz-Novikov cohomology, and computes these groups for specific nilmanifold and solvmanifold examples.

Contribution

It extends spectral sequence techniques to study primitive cohomology in locally conformal symplectic geometry and computes these groups for particular classes of manifolds.

Findings

Primitive cohomology groups are computed for certain nilmanifolds and solvmanifolds.

L.c.s. solvmanifold is shown to be a mapping torus of a non-isotopic contactomorphism.

Spectral sequences relate primitive cohomology to Lichnerowicz-Novikov cohomology.

Abstract

In this note we introduce primitive cohomology groups of locally conformal symplectic manifolds $(M^{2n}, \omega, \theta)$. We study the relation between the primitive cohomology groups and the Lichnerowicz-Novikov cohomology groups of $(M^{2n}, \omega, \theta)$, using and extending the technique of spectral sequences developed by Di Pietro and Vinogradov for symplectic manifolds. We discuss related results by many peoples, e.g. Bouche, Lychagin, Rumin, Tseng-Yau, in light of our spectral sequences. We calculate the primitive cohomology groups of a $(2n+2)$-dimensional locally conformal symplectic nilmanifold as well as those of a l.c.s. solvmanifold. We show that the l.c.s. solvmanifold is a mapping torus of a contactomorphism, which is not isotopic to the identity.