Symplectic harmonicity and generalized coeffective cohomologies
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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 Tsai, Tseng and Yau. We construct closed (simply connected) manifolds endowed with a family of symplectic forms \(\omega _t\) 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.
KeywordsSymplectic Hodge theory Coeffective cohomology Filtered and primitive cohomologies Lefschetz map
Mathematics Subject Classification53D05 53D35 57R17
This work has been partially supported by the Projects MTM2017-85649-P (AEI/FEDER, UE), and E22-17R “Algebra y Geometría” (Gobierno de Aragón/FEDER). We thank the referee for useful suggestions and for comments on the relations of the generalized coeffective complex and the filtered complex of Tsai, Tseng and Yau.
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