Symplectic harmonicity and generalized coeffective cohomologies

TL;DR

This paper explores the relationships between symplectically harmonic and coeffective cohomologies, introduces a generalized coeffective cohomology, and constructs examples demonstrating the variability of these cohomologies across different symplectic forms.

Contribution

It generalizes coeffective cohomology to include filtered cohomologies and provides explicit examples of manifolds with variable symplectic cohomology dimensions.

Findings

Cohomology dimensions vary with symplectic form parameter t.

Complete cohomological analysis for 6D symplectic nilmanifolds.

Concrete examples on 8D solvmanifolds and higher-dimensional nilmanifolds.

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 $\omega_t$ 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.