---
tag: 7SZF58P6Z7
kind: definition
name: Class VII surface
msc: [32J15]
defines: Y4NSSJT4RC
uses: [JSGWAN2RJV, ZW00Q0SAGR, ST71ZV3FES]
sources:
  - url: https://en.wikipedia.org/wiki/Surface_of_class_VII
---

## Definition

Let $X$ be a compact, connected [[complex manifold|JSGWAN2RJV]] of dimension $2$.
Then $X$ is a *surface of class VII* if:

1. its [[Kodaira dimension|ZW00Q0SAGR]] is $\kappa(X) = -\infty$, and
2. its first [[Betti number|ST71ZV3FES]] is $b_1(X) = 1$.

A class VII surface is in *class $\mathrm{VII}_0$* if it is minimal, that is, it contains no smooth rational curve of self-intersection $-1$.

## Remarks

Example: a [[Hopf surface|PNFNGBXRM3]] is a class VII surface.
For 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.
[[Inoue surfaces|GPSVJK3R4X]] are class VII surfaces with $b_2 = 0$ and no curves.

Non-example: $\mathbb{CP}^2$ has $\kappa = -\infty$, but $b_1 = 0$, so clause 2 fails; it is a rational surface.

Since $b_1$ is odd, a class VII surface is not Kähler, and in particular not projective.
Blowing 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.
A theorem of Bogomolov states that every class $\mathrm{VII}_0$ surface with $b_2 = 0$ is a Hopf surface or an Inoue surface.
Li, Yau and Zheng, and later Teleman, gave complete proofs.
For $b_2 > 0$ the classification is open; the global spherical shell conjecture predicts that every class $\mathrm{VII}_0$ surface with $b_2 > 0$ contains a global spherical shell.