Cohomologies of locally conformally symplectic manifolds and solvmanifolds
Let \(X\) be a compact differentiable manifold and let \(\Omega\) be a locally conformal symplectic (lcs) form on \(X\), let \(\theta\) be the Lee form and let \(d_{\theta}:=d- (\theta \wedge-)\) be the twisted differential. Lcs cohomologies on \(X\) are introduced and studied. Precisely, a bi-differential graded vector space structure is introduced and its associated cohomologies are investigated: Morse-Novikov cohomology and its lcs-dual, lcs Bott-Chern and Aeppli cohomologies. The elliptic Hodge theory, the Poincaré duality, and the hard Lefschetz condition are studied. Moreover explicit examples are described on nilmanifolds, solvmanifolds and Oeljeklaus-Toma manifolds. It is proved that Inoue surfaces of type \(S^0\) and in general Oeljeklaus-Toma manifolds with one complex plane satisfy the Mostow property.
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