  Complex manifolds with compatible metric have a naturally defined subspace of
harmonic differential forms that satisfy Serre, Hodge, and conjugation duality,
as well as hard Lefschetz duality. This last property follows from a
representation of $sl(2,\mathbb{C})$, generalizing the well known structure on
the harmonic forms of compact K\"ahler manifolds. Some topological implications
are deduced.
