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  "body": "\n## Summary\n\nThe $\\bar\\partial$ operator, read \"d-bar\", sends a [[$(p, q)$-form|DC81TCR98Y]] to a $(p, q+1)$-form by differentiating its coefficients in the conjugate variables $\\bar z_1, \\dots, \\bar z_n$.\nOn functions it is $\\bar\\partial f = \\sum_j \\frac{\\partial f}{\\partial \\bar z_j} \\, d\\bar z_j$, and $\\bar\\partial f = 0$ is exactly the system of [[Cauchy–Riemann equations in several variables|9PT772R94A]].\nIt satisfies $\\bar\\partial \\circ \\bar\\partial = 0$ and splits the [[exterior derivative|0ZHSTAG8ZY]] as $d = \\partial + \\bar\\partial$.\nIt generalises the one-variable [[Cauchy–Riemann operator|PPSSTHTEDE]], and it is the basic operator of Dolbeault cohomology and of the $L^2$ methods of several complex variables.\n\n## Explanations\n\n### L1 human\n\nThe exterior derivative $d$ measures how a function or form changes in all real directions.\nUsing complex coordinates, this change splits into a part along $z_1, \\dots, z_n$ and a part along $\\bar z_1, \\dots, \\bar z_n$.\nThe $\\bar\\partial$ operator keeps only the second part.\nA function is holomorphic exactly when this part is zero.\nFor example, $\\bar\\partial(z_1 \\bar z_2) = z_1 \\, d\\bar z_2$, which is not zero, so $z_1 \\bar z_2$ is not holomorphic.\n",
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