= Hard Lefschetz isomorphism on Dolbeault cohomology
{title2=$L^{n-p-q}:H_{\bar\partial}^{p,q}\cong H_{\bar\partial}^{n-q,n-p}\quad(p+q\le n)$}
On a compact <Kähler manifold>, the <Lefschetz operator preserves harmonic forms>, so the indicated power acts on harmonic representatives of <Dolbeault cohomology>. It is injective by <injectivity of powers of the Lefschetz operator>; <Hodge symmetry> and <Hodge duality> make its finite-dimensional source and target have equal dimension, so it is an isomorphism.
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