Under a spectral gap condition comparing two Laplacians, harmonic forms on closed almost Kähler manifolds decompose by type, satisfy Hard Lefschetz, and in dimension 4 force integrability when b+ ≥ 2.
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Hodge decomposition and Hard Lefschetz Condition on almost K\"{a}hler manifolds
Under a spectral gap condition comparing two Laplacians, harmonic forms on closed almost Kähler manifolds decompose by type, satisfy Hard Lefschetz, and in dimension 4 force integrability when b+ ≥ 2.