Dolbeault Hodge decomposition on a compact Hermitian manifold

For a compact Hermitian manifold, elliptic theory gives the orthogonal decompositionThe harmonic space ${\mathcal{H}}_{\bar{\partial }}^{p,q}=\ker {\Delta }_{\bar{\partial }}$ is finite-dimensional and represents Dolbeault cohomology.

On a compact Kähler manifold, a pure-type differential form that is $d$-closed and is either $\partial$-exact or $\bar{\partial }$-exact is $\partial \bar{\partial }$-exact. In particular, if $\eta$ is $\bar{\partial }$-exact and $\partial \eta =0$, thenfor a form $\varphi$ of bidegree one lower in each component.

On a compact Kähler manifold, if a differential form $\alpha$ is $d$-closed and $d^c$-exact, then it is $d{d}^c$-exact: there is a form $\beta$ two degrees lower such that $\alpha =d{d}^c\beta$. This real-form statement is equivalent, after decomposing by type, to the ddbar lemma.