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  "body": "\n## Summary\n\nA class VII surface is a compact complex surface with [[Kodaira dimension|ZW00Q0SAGR]] $-\\infty$ and first [[Betti number|ST71ZV3FES]] $1$.\nThe name comes from Kodaira's list of classes in the [[Enriques–Kodaira classification|1GVF19KFAK]].\nThese surfaces are never Kähler, since $b_1$ is odd.\nThey form the part of the classification of compact complex surfaces that is still not complete.\n[[Hopf surfaces|PNFNGBXRM3]] and [[Inoue surfaces|GPSVJK3R4X]] are the minimal examples with second Betti number $0$.\n\n## Explanations\n\n### L1 human\n\nCompact complex surfaces are sorted into classes by a few numbers, such as how many independent holomorphic forms of top degree their powers carry and the number of independent loops.\nClass VII collects the surfaces with no such forms and with a single independent loop.\nThe simplest one is the product of a circle and a three-dimensional sphere, given a complex structure as a Hopf surface.\nSuch surfaces cannot be described by polynomial equations, and many questions about them are still open.\n",
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