Cohomology theories on locally conformal symplectic manifolds

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.

Keywords

locally conformal symplectic manifold, Lichnerowicz-Novikov cohomology, primitive cohomology, spectral sequence

2010 Mathematics Subject Classification

53D35, 55Txx, 57R17

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Published 12 February 2015