Asian Journal of Mathematics

Volume 19 (2015)

Number 1

Cohomology theories on locally conformal symplectic manifolds

Pages: 45 – 82



Hông Vân Lê (Institute of Mathematics of ASCR, Praha, Czech Republic)

Jiri Vanžura (Institute of Mathematics of ASCR, Brno, Czech Republic)


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.


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

2010 Mathematics Subject Classification

53D35, 55Txx, 57R17

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