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  "body": "\n## Definition\n\nLet $M$ be a smooth [[CR manifold|ZC78JKZX9S]] with CR bundle $T^{1,0}M \\subset \\C TM$, and let $T^{0,1}M = \\overline{T^{1,0}M}$.\nFor $q \\ge 0$, a $(0,q)$-form on $M$ is a smooth section $\\varphi$ of the bundle $\\Lambda^q (T^{0,1}M)^*$.\nEquivalently, $\\varphi$ assigns a smooth function to $q$ smooth sections of $T^{0,1}M$, and it is $C^\\infty(M)$-linear and alternating.\nWrite $\\mathcal{E}^{0,q}(M)$ for the space of $(0,q)$-forms.\nLet $\\bar Z_0, \\dots, \\bar Z_q$ be sections of $T^{0,1}M$.\nA hat marks an omitted argument, and $[\\cdot, \\cdot]$ is the [[Lie bracket of vector fields|68FGQEBD6J]].\nThe tangential Cauchy–Riemann operator $\\bar\\partial_b \\colon \\mathcal{E}^{0,q}(M) \\to \\mathcal{E}^{0,q+1}(M)$ is given by\n$$\n\\begin{aligned}\n(\\bar\\partial_b \\varphi)(\\bar Z_0, \\dots, \\bar Z_q) ={} \u0026 \\sum_{j=0}^{q} (-1)^j\\, \\bar Z_j \\bigl(\\varphi(\\bar Z_0, \\dots, \\widehat{\\bar Z_j}, \\dots, \\bar Z_q)\\bigr) \\\\\n\u0026 + \\sum_{0 \\le j \u003c k \\le q} (-1)^{j+k}\\, \\varphi\\bigl([\\bar Z_j, \\bar Z_k], \\bar Z_0, \\dots, \\widehat{\\bar Z_j}, \\dots, \\widehat{\\bar Z_k}, \\dots, \\bar Z_q\\bigr)\n\\end{aligned}\n$$\nIt satisfies $\\bar\\partial_b \\circ \\bar\\partial_b = 0$.\nThe *Kohn–Rossi cohomology* groups of $M$ are\n$$H^{0,q}_b(M) = \\frac{\\ker\\bigl(\\bar\\partial_b \\colon \\mathcal{E}^{0,q}(M) \\to \\mathcal{E}^{0,q+1}(M)\\bigr)}{\\operatorname{im}\\bigl(\\bar\\partial_b \\colon \\mathcal{E}^{0,q-1}(M) \\to \\mathcal{E}^{0,q}(M)\\bigr)}, \\qquad q \\ge 0,$$\nwith $\\mathcal{E}^{0,-1}(M) = 0$.\n\n## Remarks\n\nThe formula makes sense because $T^{0,1}M$ is closed under the Lie bracket, which is the integrability condition in the definition of a CR manifold; this also gives $\\bar\\partial_b^2 = 0$.\nIt is the formula for the exterior derivative of a [[differential form|0E3G0GRPZ6]], applied only to vectors in $T^{0,1}M$.\nFor $q = 0$, $\\bar\\partial_b u$ is the restriction of $du$ to $T^{0,1}M$, so $H^{0,0}_b(M)$ is the space of smooth CR functions on $M$.\nExample: on $M = \\C^n$ with $T^{1,0}M$ the usual holomorphic tangent bundle, $\\bar\\partial_b = \\bar\\partial$, and $H^{0,q}_b(\\C^n)$ is the Dolbeault cohomology $H^{0,q}(\\C^n)$, which is $0$ for $q \\ge 1$.\nNon-example: on the unit sphere $S^3 \\subset \\C^2$, with $\\bar L = z_2 \\partial / \\partial \\bar z_1 - z_1 \\partial / \\partial \\bar z_2$, the function $u = \\bar z_1$ satisfies $\\bar\\partial_b u (\\bar L) = \\bar L \\bar z_1 = z_2 \\ne 0$, so it is not a cocycle and gives no element of $H^{0,0}_b(S^3)$.\nMore generally one defines groups $H^{p,q}_b(M)$ using forms of type $(p,q)$.\nFor a compact strongly pseudoconvex CR manifold of hypersurface type and real dimension $2n + 1$, Kohn proved that $H^{0,q}_b(M)$ is finite-dimensional for $1 \\le q \\le n - 1$.\n",
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