 Bott-Chern cohomology of a compact Kähler manifold

For a compact Kähler manifold, the natural map $H_{BC}^{p,q}\to H_{\bar{\partial }}^{p,q}$ is an isomorphism. The Kähler Laplacian identity makes the harmonic representative of every Dolbeault cohomology class $d$-closed, giving surjectivity, and the ddbar lemma gives injectivity.