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  "body": "\n## Definition\n\nLet $X$ be a [[complex manifold|JSGWAN2RJV]] of complex dimension $n$, and let $q$ be an integer with $1 \\le q \\le n$.\nLet $\\varphi \\colon X \\to \\R$ be a [[smooth|0V33BNKZ9S]] function.\nIt is *strongly $q$-convex* at a point $x$ if its [[Levi form|80MTNXWW80]] at $x$, in local holomorphic coordinates, has at least $n - q + 1$ positive eigenvalues.\nThe manifold $X$ is *$q$-convex* if there are a smooth function $\\varphi \\colon X \\to \\R$ and a [[compact set|0R6EWFKRCF]] $K \\subseteq X$ such that:\n\n1. $\\varphi$ is an exhaustion: the set $\\set{x \\in X : \\varphi(x) \u003c c}$ is relatively compact in $X$ for every $c \\in \\R$.\n2. $\\varphi$ is strongly $q$-convex at every point of $X \\setminus K$.\n\nThe manifold $X$ is *$q$-complete* if this holds with $K = \\emptyset$.\n\n## Remarks\n\nThis is the convention of Andreotti and Grauert; some authors shift the index by one.\nThe number of positive eigenvalues of the Levi form does not depend on the choice of local coordinates.\nExample: $\\C^n$ is $1$-complete with $\\varphi(z) = \\abs{z}^2$, whose Levi form $\\abs{w}^2$ has $n$ positive eigenvalues.\nNon-example: a compact complex manifold of positive dimension is $q$-convex with $K = X$, but it is not $1$-complete.\nIndeed a strongly $1$-convex function is strictly plurisubharmonic, so it has no maximum, while every continuous function on a compact manifold has one.\nThe $1$-complete manifolds are exactly the Stein manifolds, and the $1$-convex manifolds are also called strongly pseudoconvex manifolds.\n",
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