Examples of non $d_{\omega}$-exact locally conformal symplectic forms

Abstract

We exhibit two three-parameter families of locally conformal symplectic forms on the solvmanifold $M_{n,k}$ considered in [1], and show, using the Hodge-de Rham theory for the Lichnerowicz cohomology that that they are not $d_{\omega}$ exact, i.e. their Lichnerowicz classes are non-trivial (Theorem 1). This has several important geometric consequences (corollary 2, 3). This also implies that the group of automorphisms of the corresponding locally conformal symplectic structures behaves much like the group of symplectic diffeomorphisms of compact symplectic manifolds. We initiate the classification of the local conformal symplectic forms in each 3-parameter family (Theorem 2, corollary 1). We also show that the first (and) third Lichnerowicz cohomology classes are non-zero (Theorem 3). We observe finally that the manifolds $M_{n,k}$ carry several interesting foliations and Poisson structures.