---
tag: Y4NSSJT4RC
kind: concept
name: Class VII surface
msc: [32J15]
specializes: [JSGWAN2RJV]
sources:
  - url: https://en.wikipedia.org/wiki/Surface_of_class_VII
same_as:
  wikidata: Q7645998
  wikipedia: Surface of class VII
---

## Summary

A class VII surface is a compact complex surface with [[Kodaira dimension|ZW00Q0SAGR]] $-\infty$ and first [[Betti number|ST71ZV3FES]] $1$.
The name comes from Kodaira's list of classes in the [[Enriques–Kodaira classification|1GVF19KFAK]].
These surfaces are never Kähler, since $b_1$ is odd.
They form the part of the classification of compact complex surfaces that is still not complete.
[[Hopf surfaces|PNFNGBXRM3]] and [[Inoue surfaces|GPSVJK3R4X]] are the minimal examples with second Betti number $0$.

## Explanations

### L1 human

Compact 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.
Class VII collects the surfaces with no such forms and with a single independent loop.
The simplest one is the product of a circle and a three-dimensional sphere, given a complex structure as a Hopf surface.
Such surfaces cannot be described by polynomial equations, and many questions about them are still open.