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  "body": "\n## Summary\n\nA $q$-convex manifold is a complex manifold with a smooth exhaustion function whose [[Levi form|80MTNXWW80]] has many positive eigenvalues outside a compact set.\nWhen this holds everywhere, the manifold is called $q$-complete.\nFor $q = 1$ one gets the strongly pseudoconvex manifolds and the Stein manifolds.\nThese classes measure how far a manifold is from being Stein, and the [[Andreotti–Grauert theorem|67RJ0S8F74]] controls their cohomology.\n\n## Explanations\n\n### L1 human\n\nA Stein manifold has an exhaustion function that is convex in every complex direction.\nA $q$-convex manifold has one that is convex at least in $n - q + 1$ independent complex directions at each point far out, where $n$ is the dimension.\nSo the larger $q$ is, the weaker the condition.\nFor example, $\\C^n$ with the origin removed is not Stein when $n \\ge 2$.\nThe function $\\abs{z}^2 + 1/\\abs{z}^2$ grows to infinity near the origin and near infinity, and its Levi form is positive in the radial direction $w = z$.\nSo $\\C^n \\setminus \\set{0}$ is $n$-complete in the convention used here.\n",
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