For a compact Kähler manifold, we say that a form is primitive if it is contaned in the kernel of the dual Lefschetz operator, or the co-Lefschetz operator. For all examples I know, a primitive form $ \omega$ is closed with respect to the $ d$ de Rham exterior derivative if and only if $ \omega$ is harmonic. I suspect that this is true for any compact Kähler manifold, but I don’t know how one would prove it.
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