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The well-known K\"ahler identities naturally extend to the non-integrablesetting. This paper deduces several geometric and topological consequences ofthese extended identities for compact almost K\"ahler manifolds. Among theseare identities of various Laplacians, generalized Hodge and Serre dualities, ageneralized hard Lefschetz duality, and a Lefschetz decomposition, all on thespace of $d$-harmonic forms of pure bidegree. There is also a generalization ofHodge Index Theorem for compact almost K\"ahler $4$-manifolds. In particular,these provide topological bounds on the dimension of the space of $d$-harmonicforms of pure bidegree, as well as several new obstructions to the existence ofa symplectic form compatible with a given almost complex structure.

Topological and geometric aspects of almost K\"ahler manifolds via harmonic theory