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  "body": "\n## Statement\n\nLet $X$ be a [[complex manifold|JSGWAN2RJV]] of complex dimension $n$, let $1 \\le q \\le n$, and let $\\mathcal{F}$ be a [[coherent analytic sheaf|DSJET02WPR]] on $X$.\n\n1. If $X$ is [[$q$-convex|VBHR4R5W7F]], then the [[sheaf cohomology|PQ6Q143ARJ]] group $H^r(X, \\mathcal{F})$ is a finite-dimensional complex vector space for every $r \\ge q$.\n2. If $X$ is $q$-complete, then $H^r(X, \\mathcal{F}) = 0$ for every $r \\ge q$.\n\n## Remarks\n\nHere $q$-convexity is taken in the convention of Andreotti and Grauert: there is a smooth exhaustion function whose Levi form has at least $n - q + 1$ positive eigenvalues outside a compact set.\nAndreotti and Grauert proved the theorem in 1962, for complex spaces as well as for manifolds.\nThe proof extends Grauert's bumping method.\nThe cohomology on a sublevel set $\\set{\\varphi \u003c c}$ is compared with the cohomology on a slightly larger one.\nLocal vanishing results near strongly $q$-convex boundary points control the restriction maps in degrees $r \\ge q$.\nThe $1$-complete manifolds are the Stein manifolds.\nFor them, part 2 says that $H^r(X, \\mathcal{F}) = 0$ for $r \\ge 1$, which is Cartan's theorem B.\nFor $q = 1$, part 1 gives Grauert's finiteness theorem for strongly pseudoconvex manifolds.\nOn $\\C^2 \\setminus \\set{0}$, which is $2$-complete but not $1$-complete, the group $H^1(X, \\mathcal{O})$ is infinite-dimensional, so the bound $r \\ge q$ cannot be lowered.\n\n## Explanations\n\n### L1 human\n\nThe cohomology groups of a coherent sheaf measure the obstructions to solving holomorphic problems on a complex manifold, such as finding a function with prescribed local behaviour.\nThe theorem says that convexity of the manifold kills these obstructions in high degrees.\nThe more convex the manifold is, the more degrees are controlled: a Stein manifold has no obstructions at all in positive degree.\n\n### L2 human\n\nThe theorem puts Cartan's theorem B and Grauert's finiteness theorem into one framework, indexed by $q$.\nApplied to the sheaf of holomorphic $p$-forms and combined with the Dolbeault isomorphism, part 2 says that on a $q$-complete manifold every $\\bar\\partial$-closed smooth $(p, r)$-form with $r \\ge q$ is $\\bar\\partial$-exact.\n",
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