Dolbeault theorem
= Dolbeault theorem
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For the sheaf $\Omega^p$ of holomorphic $p$-forms on a <complex manifold>, the <Dolbeault-Poincaré lemma> makes
$$
0\longrightarrow\Omega^p\longrightarrow\mathcal A^{p,0}\xrightarrow{\bar\partial}\mathcal A^{p,1}\xrightarrow{\bar\partial}\cdots
$$
a resolution by <fine sheaves>. Taking <global sections> therefore gives the natural isomorphism
$$
H^q(X,\Omega^p)\cong H_{\bar\partial}^{p,q}(X).
$$