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  "body": "\n## Statement\n\nLet $X$ be a [[complex manifold|JSGWAN2RJV]], let $p, q \\ge 0$ be integers, and let $\\Omega^p_X$ be the sheaf of holomorphic $p$-forms on $X$.\nThen there is a natural isomorphism of complex vector spaces\n$$H^q(X, \\Omega^p_X) \\cong H^{p,q}_{\\bar\\partial}(X)$$\nbetween the [[sheaf cohomology|PQ6Q143ARJ]] of $\\Omega^p_X$ and the [[Dolbeault cohomology|Y3900T2EY5]] of $X$.\nMore generally, let $E$ be a holomorphic vector bundle on $X$.\nThen $H^q(X, \\Omega^p_X(E))$ is isomorphic to the cohomology of the complex of smooth $E$-valued $(p,\\cdot)$-forms with the operator $\\bar\\partial_E$.\n\n## Remarks\n\nLet $\\mathcal{A}^{p,q}$ be the sheaf of smooth $(p,q)$-forms.\nThe [[Dolbeault–Grothendieck lemma|5858M5PGN6]] shows that the sequence of sheaves\n$$0 \\to \\Omega^p_X \\to \\mathcal{A}^{p,0} \\xrightarrow{\\bar\\partial} \\mathcal{A}^{p,1} \\xrightarrow{\\bar\\partial} \\mathcal{A}^{p,2} \\to \\cdots$$\nis exact.\nThe sheaves $\\mathcal{A}^{p,q}$ are fine, because they are modules over the sheaf of smooth functions, which has partitions of unity, so they have no higher cohomology.\nThe cohomology of $\\Omega^p_X$ is therefore the cohomology of the complex of global sections, which is the Dolbeault complex.\nExample: on a compact Riemann surface $X$ of genus $g$, $H^1(X, \\mathcal{O}_X) \\cong H^{0,1}_{\\bar\\partial}(X)$, and both have dimension $g$.\nFor $p = 0$, the theorem identifies the cohomology of the structure sheaf $\\mathcal{O}_X = \\Omega^0_X$ with the obstruction groups to solving $\\bar\\partial u = f$.\n\n## Explanations\n\n### L1 human\n\nSheaf cohomology is defined abstractly and is hard to compute directly.\nThe Dolbeault isomorphism says that the cohomology of holomorphic forms can be computed with differential forms and the operator $\\bar\\partial$.\nThis turns algebraic questions into questions about a partial differential equation.\n\n### L2 human\n\nThe isomorphism is the complex analogue of de Rham's theorem, which computes the cohomology of the constant sheaf with smooth forms and the operator $d$.\nIt lets one prove vanishing theorems for sheaf cohomology with analytic tools, such as Hörmander's $L^2$ estimates or harmonic forms.\nOn compact Kähler manifolds, Hodge theory represents each class by a harmonic form, and the isomorphism then gives the Hodge decomposition $H^k(X, \\C) \\cong \\bigoplus_{p+q=k} H^q(X, \\Omega^p_X)$.\n",
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