Hard Lefschetz isomorphism on Dolbeault cohomology ($L^{n-p-q}:H_{\bar{\partial }}^{p,q}\cong H_{\bar{\partial }}^{n-q,n-p}(p+q\le n)$)

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.