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  "body": "\n## Summary\n\nKohn–Rossi cohomology is the cohomology of the tangential Cauchy–Riemann complex $\\bar\\partial_b$ on a [[CR manifold|ZC78JKZX9S]].\nIt plays for CR manifolds the role that [[Dolbeault cohomology|Y3900T2EY5]] plays for complex manifolds.\nIts degree $0$ group is the space of smooth [[CR functions|JFKYYK2CEB]], and the higher groups measure the obstruction to solving the equation $\\bar\\partial_b u = f$.\n\n## Explanations\n\n### L1 human\n\nOn a complex manifold one studies the equation $\\bar\\partial u = f$, and Dolbeault cohomology records when it cannot be solved.\nA CR manifold, such as a real hypersurface in $\\C^n$, has only part of the complex directions, so only the tangential part $\\bar\\partial_b$ of $\\bar\\partial$ makes sense on it.\nKohn–Rossi cohomology records when $\\bar\\partial_b u = f$ cannot be solved, given the necessary condition $\\bar\\partial_b f = 0$.\nTake the unit sphere $S^{2n-1}$ in $\\C^n$ with $n \\ge 2$.\nIts degree $0$ group consists of the boundary values of holomorphic functions on the ball that are smooth up to the boundary.\n",
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