Symplectic harmonicity and generalized coeffective cohomologies

Abstract

Relations between the symplectically harmonic cohomology and the coeffective cohomology of a symplectic manifold are obtained. This is achieved through a generalization of the latter, which in addition allows us to provide a coeffective version of the filtered cohomologies introduced by C.-J. Tsai, L.-S. Tseng and S.-T. Yau. We construct closed (simply connected) manifolds endowed with a family of symplectic forms ωt such that the dimensions of these symplectic cohomology groups vary with respect to t. A complete study of these cohomologies is given for 6-dimensional symplectic nilmanifolds, and concrete examples with special cohomological properties are obtained on an 8-dimensional solvmanifold and on 2-step nilmanifolds in higher dimensions.

Harmonic almost complex structures on almost abelian Lie groups and solvmanifolds

“…We recall that a solvmanifold is a compact quotient M Γ = Γ\G of a simply connected solvable Lie group G by a discrete subgroup Γ. They constitute a fruitful and interesting source of examples in (almost) Hermitian, symplectic and G 2 geometry, among others (see for instance [13,14,23,28,32]). Indeed, when studying the properties of geometric structures on M Γ induced by a left invariant one on the Lie group G, we can work at the Lie algebra level and consider the associated linear objects.…”