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  "body": "\n## Summary\n\nDolbeault cohomology is the cohomology of the complex of smooth [[(p,q)-forms|DC81TCR98Y]] on a [[complex manifold|JSGWAN2RJV]] with the [[dbar operator|M1VKCW3RFW]] as differential.\nIt measures to what extent the equation $\\bar\\partial u = f$ fails to be solvable for $\\bar\\partial$-closed data $f$.\nBy the [[Dolbeault isomorphism|G2HC1SFQ4P]] it computes the sheaf cohomology of the sheaves of holomorphic $p$-forms.\nOn compact Kähler manifolds it gives the Hodge decomposition of de Rham cohomology.\n\n## Explanations\n\n### L1 human\n\nOn a complex manifold, a differential form splits into pieces of type $(p,q)$: $p$ holomorphic differentials $dz_j$ and $q$ antiholomorphic differentials $d\\bar z_k$.\nThe operator $\\bar\\partial$ differentiates in the antiholomorphic directions and squares to zero.\nDolbeault cohomology is the space of $\\bar\\partial$-closed forms modulo $\\bar\\partial$-exact forms.\nFor example, $H^{0,0}$ is the space of holomorphic functions, since a function with $\\bar\\partial f = 0$ is holomorphic.\nOn a polydisc in $\\C^n$ all groups $H^{p,q}$ with $q \\ge 1$ vanish, while on a compact Riemann surface of genus $g$ the group $H^{0,1}$ has dimension $g$.\n",
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