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  "body": "\n## Definition\n\nLet $X$ be a compact, connected [[complex manifold|JSGWAN2RJV]] of dimension $2$.\nThen $X$ is a *surface of class VII* if:\n\n1. its [[Kodaira dimension|ZW00Q0SAGR]] is $\\kappa(X) = -\\infty$, and\n2. its first [[Betti number|ST71ZV3FES]] is $b_1(X) = 1$.\n\nA class VII surface is in *class $\\mathrm{VII}_0$* if it is minimal, that is, it contains no smooth rational curve of self-intersection $-1$.\n\n## Remarks\n\nExample: a [[Hopf surface|PNFNGBXRM3]] is a class VII surface.\nFor the diagonal Hopf surface, $b_1 = 1$ because it is diffeomorphic to $S^1 \\times S^3$, and no power of the canonical bundle has a nonzero holomorphic section.\n[[Inoue surfaces|GPSVJK3R4X]] are class VII surfaces with $b_2 = 0$ and no curves.\n\nNon-example: $\\mathbb{CP}^2$ has $\\kappa = -\\infty$, but $b_1 = 0$, so clause 2 fails; it is a rational surface.\n\nSince $b_1$ is odd, a class VII surface is not Kähler, and in particular not projective.\nBlowing up a point keeps $\\kappa$ and $b_1$ and raises $b_2$ by $1$, so every class VII surface is a blow-up of a minimal one.\nA theorem of Bogomolov states that every class $\\mathrm{VII}_0$ surface with $b_2 = 0$ is a Hopf surface or an Inoue surface.\nLi, Yau and Zheng, and later Teleman, gave complete proofs.\nFor $b_2 \u003e 0$ the classification is open; the global spherical shell conjecture predicts that every class $\\mathrm{VII}_0$ surface with $b_2 \u003e 0$ contains a global spherical shell.\n",
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