---
tag: PNFNGBXRM3
kind: example
name: Hopf surface
msc: [32J15]
instance_of: [Y4NSSJT4RC]
uses: [JSGWAN2RJV, Y1YPE6PM52, ZW00Q0SAGR, ST71ZV3FES]
sources:
  - url: https://en.wikipedia.org/wiki/Hopf_surface
same_as:
  wikidata: Q5900512
  wikipedia: Hopf surface
---

## Example

A *Hopf surface* is a compact complex surface whose [[universal cover|Y1YPE6PM52]] is biholomorphic to $\C^2 \setminus \set{0}$.
It is *primary* if its fundamental group is infinite cyclic.

Example: let $0 < \abs{\lambda} < 1$ and $X = (\C^2 \setminus \set{0}) / \langle g \rangle$ with $g(z, w) = (\lambda z, \lambda w)$, the [[Hopf manifold|Z8R2D33WSM]] of dimension $2$.
Since $\C^2 \setminus \set{0}$ is homotopy equivalent to $S^3$, it is simply connected, so it is the universal cover of $X$, and $\pi_1(X) \cong \Z$.

Check that $X$ is a [[class VII surface|Y4NSSJT4RC]].
Its first [[Betti number|ST71ZV3FES]] is $1$, since $X$ is diffeomorphic to $S^1 \times S^3$.
Its [[Kodaira dimension|ZW00Q0SAGR]] is $-\infty$.
A section of $K_X^{\otimes m}$ with $m \ge 1$ pulls back to $h(z, w)\,(dz \wedge dw)^{\otimes m}$ with $h$ holomorphic on $\C^2 \setminus \set{0}$, hence, by Hartogs's extension theorem, on $\C^2$.
Invariance under $g$ gives $h(\lambda z, \lambda w)\, \lambda^{2m} = h(z, w)$, so each Taylor coefficient $c_{jk}$ of $h$ satisfies $c_{jk} \lambda^{j + k + 2m} = c_{jk}$; since $\abs{\lambda} < 1$, all $c_{jk}$ vanish and $h = 0$.

## Remarks

Kodaira showed that every primary Hopf surface is the quotient of $\C^2 \setminus \set{0}$ by the infinite cyclic group generated by a holomorphic contraction that fixes $0$.
Every Hopf surface has $b_1 = 1$ and $b_2 = 0$, so it is not Kähler.
The function $z/w$ on the example above is invariant under $g$, so it gives a nonconstant meromorphic function on $X$.
It also gives a map to $\mathbb{CP}^1$ whose fibres are elliptic curves, and this $X$ has [[algebraic dimension|F0GF0JEZTG]] $1$.

## Explanations

### L1 human

A Hopf surface is a compact complex surface obtained from $\C^2$ minus the origin by gluing each point to its images under a contraction toward the origin.
Topologically it is usually the product of a circle and a three-dimensional sphere.
It is the first and simplest example of a compact complex surface that is not Kähler, and so not algebraic.