= Solution
The <Hodge decomposition theorem for compact Kähler manifolds> gives a unique <harmonic differential form> representative of every complex <de Rham cohomology> class, with a decomposition into harmonic pure-type components. Consequently
$$
\boxed{H^k_{dR}(X;\mathbb C)=\bigoplus_{p+q=k}H_{\bar\partial}^{p,q}(X).}
$$
Equivalently, the <Dolbeault Hodge decomposition on a compact Hermitian manifold> is
$$
\Omega^{p,q}=\mathcal H^{p,q}\oplus\bar\partial\Omega^{p,q-1}\oplus\bar\partial^*\Omega^{p,q+1}.
$$
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