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  "body": "\n## Summary\n\nA $(p, q)$-form on an open subset of $\\C^n$, or on a complex manifold, is a complex-valued [[differential form|0E3G0GRPZ6]] of a special type.\nEach of its terms contains $p$ of the differentials $dz_j$ and $q$ of the conjugate differentials $d\\bar z_j$.\nEvery complex differential form of degree $k$ splits in a unique way as a sum of $(p, q)$-forms with $p + q = k$.\nThis splitting does not depend on the choice of holomorphic coordinates.\nIt is the setting of the [[$\\bar\\partial$ operator|M1VKCW3RFW]] and of Dolbeault cohomology.\n\n## Explanations\n\n### L1 human\n\nIn $\\C$, the differential $dx$ of the real part and $dy$ of the imaginary part can be traded for $dz = dx + i\\,dy$ and $d\\bar z = dx - i\\,dy$.\nA $(p, q)$-form is a form whose terms each contain exactly $p$ factors of the kind $dz_j$ and $q$ of the kind $d\\bar z_j$.\nFor example, $dz_1 \\wedge d\\bar z_2$ in $\\C^2$ is a $(1, 1)$-form.\nThe form $dx_1 = \\tfrac12 (dz_1 + d\\bar z_1)$ is a sum of a $(1, 0)$-form and a $(0, 1)$-form, so it has no single type.\n",
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  "name": "(p,q)-form",
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  "url": "https://tamnd.github.io/math2/wiki/p-q-form/",
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