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  "body": "\n## Definition\n\nLet $U \\subseteq \\C^n$ be an open set, with coordinates $z_1, \\dots, z_n$.\nFor a smooth function $f \\colon U \\to \\C$, put\n$$\\bar\\partial f = \\sum_{k=1}^n \\frac{\\partial f}{\\partial \\bar z_k} \\, d\\bar z_k,$$\nwhere $\\partial / \\partial \\bar z_k$ is the [[Wirtinger derivative|WMXDVF36ZT]] in the variable $z_k$.\nFor a [[$(p, q)$-form|DC81TCR98Y]] $\\omega = \\sum_{I, J} \\omega_{IJ} \\, dz_I \\wedge d\\bar z_J$ on $U$, the *$\\bar\\partial$ operator* gives the $(p, q+1)$-form\n$$\\bar\\partial \\omega = \\sum_{I, J} \\bar\\partial \\omega_{IJ} \\wedge dz_I \\wedge d\\bar z_J = \\sum_{I, J} \\sum_{k=1}^n \\frac{\\partial \\omega_{IJ}}{\\partial \\bar z_k} \\, d\\bar z_k \\wedge dz_I \\wedge d\\bar z_J.$$\nIt is extended to all complex forms by additivity over the types.\n\n## Remarks\n\nThe operator $\\partial$ is defined in the same way with $\\partial / \\partial z_k$ and $dz_k$; it sends $(p, q)$-forms to $(p+1, q)$-forms.\nOn forms on $U$, the [[exterior derivative|0ZHSTAG8ZY]] is $d = \\partial + \\bar\\partial$, and $\\bar\\partial \\circ \\bar\\partial = 0$.\nIt obeys the Leibniz rule $\\bar\\partial(\\alpha \\wedge \\beta) = \\bar\\partial \\alpha \\wedge \\beta + (-1)^{k} \\alpha \\wedge \\bar\\partial \\beta$ for $\\alpha$ of degree $k$.\nExample: for $f(z_1, z_2) = \\abs{z_1}^2 + z_2$, $\\partial f / \\partial \\bar z_1 = z_1$ and $\\partial f / \\partial \\bar z_2 = 0$, so $\\bar\\partial f = z_1 \\, d\\bar z_1$.\nNon-example: $\\bar\\partial$ is not the exterior derivative, since $\\bar\\partial z_1 = 0$ while $d z_1 = dz_1 \\ne 0$; the $(1, 0)$ part of $d$ is dropped.\nA $C^1$ function $f$ satisfies $\\bar\\partial f = 0$ on $U$ if and only if it is holomorphic on $U$.\nBecause holomorphic changes of coordinates preserve types, $\\bar\\partial$ is well defined on complex manifolds.\n",
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