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  "body": "\n## Summary\n\nThe Dolbeault operator $\\bar\\partial$ on a [[complex manifold|JSGWAN2RJV]] is a part of the [[exterior derivative|0ZHSTAG8ZY]].\nIt is the part that raises the antiholomorphic degree of a complex [[differential form|0E3G0GRPZ6]] by one.\nA function is [[holomorphic|0NAHXR1FRH]] exactly when $\\bar\\partial f = 0$, so $\\bar\\partial$ is the Cauchy–Riemann operator of complex geometry.\nIt satisfies $\\bar\\partial^2 = 0$, and the cohomology of the resulting Dolbeault complex computes the sheaf cohomology of holomorphic forms.\nThe Dolbeault complex is an [[elliptic complex|M8GMH8GQR6]], and its index is the holomorphic Euler characteristic given by the Riemann–Roch theorems.\n\n## Explanations\n\n### L1 human\n\nIn one complex variable $z = x + iy$, a function is holomorphic when it satisfies the Cauchy–Riemann equations.\nThese can be written as one equation $\\partial f / \\partial \\bar z = 0$, where $\\partial / \\partial \\bar z = \\frac{1}{2}(\\partial_x + i \\partial_y)$.\nThe Dolbeault operator packages this derivative as $\\bar\\partial f = \\frac{\\partial f}{\\partial \\bar z} \\, d\\bar z$, and extends it to forms and to several variables.\nFor example $\\bar\\partial z = 0$, since $z$ is holomorphic, while $\\bar\\partial \\bar z = d\\bar z \\ne 0$.\n",
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