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  "body": "\n## Definition\n\nLet $X$ be a [[complex manifold|JSGWAN2RJV]] of complex dimension $n$.\nFor integers $p, q \\ge 0$, let $\\Omega^{p,q}(X)$ be the space of smooth complex [[differential forms|0E3G0GRPZ6]] of type $(p, q)$.\nIn each holomorphic chart $(z_1, \\ldots, z_n)$ they are the sums $\\sum_{\\abs{I} = p, \\abs{J} = q} f_{IJ} \\, dz_I \\wedge d\\bar z_J$ with smooth functions $f_{IJ}$.\nThe [[exterior derivative|0ZHSTAG8ZY]] maps $\\Omega^{p,q}(X)$ into $\\Omega^{p+1,q}(X) \\oplus \\Omega^{p,q+1}(X)$.\nThe *Dolbeault operator* $\\bar\\partial \\colon \\Omega^{p,q}(X) \\to \\Omega^{p,q+1}(X)$ is $d$ followed by the projection onto $\\Omega^{p,q+1}(X)$.\n\n## Remarks\n\nIn a chart, with the [[Wirtinger derivatives|WMXDVF36ZT]] $\\partial / \\partial \\bar z_k = \\frac{1}{2}(\\partial / \\partial x_k + i \\, \\partial / \\partial y_k)$ for $z_k = x_k + i y_k$,\n$$\\bar\\partial \\Bigl(\\sum_{I, J} f_{IJ} \\, dz_I \\wedge d\\bar z_J\\Bigr) = \\sum_{I, J} \\sum_{k=1}^n \\frac{\\partial f_{IJ}}{\\partial \\bar z_k} \\, d\\bar z_k \\wedge dz_I \\wedge d\\bar z_J.$$\nThe other component of $d$ is written $\\partial$, so $d = \\partial + \\bar\\partial$, and $d^2 = 0$ gives $\\bar\\partial^2 = 0$, $\\partial^2 = 0$ and $\\partial \\bar\\partial + \\bar\\partial \\partial = 0$.\nThe splitting of $d$ into two components uses that the complex structure comes from holomorphic charts.\nThe cohomology of $\\bar\\partial$ on $\\Omega^{p, \\bullet}(X)$ is the Dolbeault cohomology $H^{p,q}_{\\bar\\partial}(X)$, and the same construction works for forms with values in a holomorphic vector bundle.\n\nExample: on $X = \\C$ with coordinate $z$, let $f(z) = z^2$.\nThen $\\partial f / \\partial \\bar z = 0$, so $\\bar\\partial f = 0$, as for every [[holomorphic function|0NAHXR1FRH]].\n\nNon-example: the function $g(z) = \\bar z$ on $\\C$ is not holomorphic, and indeed $\\bar\\partial g = \\frac{\\partial \\bar z}{\\partial \\bar z} \\, d\\bar z = d\\bar z \\ne 0$.\n\n## Explanations\n\n### L1 human\n\nThe Dolbeault operator takes the derivative of a form in the antiholomorphic directions only, and holomorphic functions are exactly the functions it kills.\n",
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