Cohomology theories on locally conformal symplectic manifolds

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.

43 pages, improved presentation, final version

References in corpus (2)

• Contact pairs and locally conformally symplectic structures
• Spectral sequence associated with a symplectic manifold

Cited by in corpus (7)

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• Locally conformal symplectic structures on Lie algebras of type I and their solvmanifolds
• The hard Lefschetz duality for locally conformally almost Kähler manifolds
• Classification of $k$-forms on ${\mathbf{R}}^n$ and the existence of associated geometry on manifolds