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 Hermitian manifold of complex dimension $n$, $C\alpha =\ast \bar{\alpha }$, with complex-linear Hodge star, is a conjugate-linear bijection from Dolbeault Laplacian harmonic $(p,q)$-forms to harmonic $(n-p,n-q)$-forms. The identities ${\bar{\partial }}^{\ast }=-\ast \partial \ast$ and ${\ast }^2=(-1{)}^{p+q}$ show that $\bar{\partial }\alpha ={\bar{\partial }}^{\ast }\alpha =0$ implies the same two equations for $C\alpha$. Its square is $(-1{)}^{p+q}$. Thus these harmonic spaces have equal complex dimension even without the Kähler condition. The natural map is conjugate-linear, not complex-linear.

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, the difference of cohomologous Kähler forms is a real, de Rham cohomology exact $(1,1)$-form. The ddbar lemma gives $\tilde{\omega }-\omega =\partial \bar{\partial }h$. Reality implies $\partial \bar{\partial }\bar{h}=-(\tilde{\omega }-\omega )$, so $f=(h-\bar{h})/(2i)$ is real and has the displayed potential equation. The difference of two real potentials is a pluriharmonic function, hence constant on each connected component. On a connected manifold the only ambiguity is one additive constant.

If $\alpha$ is a $d$-closed $(p,q)$-form on a compact Kähler manifold, then $\alpha$ is ddbar lemma exact if and only if it is orthogonal to every harmonic differential form of type $(p,q)$. The forward implication follows by integration by parts. For the reverse implication, Dolbeault Hodge decomposition first makes $\alpha$ $\bar{\partial }$-exact, and the ddbar lemma then makes it $\partial \bar{\partial }$-exact.

For a $d$-closed pure-type form $\alpha$ orthogonal to harmonic forms on a compact Kähler manifold, let $G_{\bar{\partial }}$ be the inverse of the Dolbeault Laplacian on the harmonic complement, zero on the harmonic space. The Kähler Laplacian identity makes it commute with $\partial$ and $\bar{\partial }$. Thus $\alpha =\bar{\partial }{\bar{\partial }}^{\ast }{G}_{\bar{\partial }}\alpha$ and $G_{\bar{\partial }}\alpha =\partial {\partial }^{\ast }{G}_{\bar{\partial }}^2\alpha$. The mixed anticommutator ${\bar{\partial }}^{\ast }\partial +\partial {\bar{\partial }}^{\ast }=0$ gives $\alpha =\partial \bar{\partial }\beta$. This is an explicit potential for the ddbar lemma, with one lower degree of each type; using the inverse of the Hodge Laplacian instead multiplies the displayed expression by four.

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.