= Complementary Dolbeault harmonic types on a Hermitian manifold
{title2=$\mathcal H^{p,q}_{\bar\partial}\cong\overline{\mathcal H^{n-p,n-q}_{\bar\partial}}$}
On a compact <Hermitian manifold> of complex dimension $n$, $C\alpha=*\bar\alpha$, 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 $\bar\partial^*=-*\partial*$ and $*^2=(-1)^{p+q}$ show that $\bar\partial\alpha=\bar\partial^*\alpha=0$ 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.
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