---
tag: GPSVJK3R4X
kind: example
name: Inoue surface
msc: [32J15]
instance_of: [Y4NSSJT4RC]
uses: [JSGWAN2RJV, X6XGGH5FPW, 0V9PMN0RBH, 0BQAA2BK3Q, ST71ZV3FES]
sources:
  - url: https://en.wikipedia.org/wiki/Inoue_surface
same_as:
  wikidata: Q6036690
  wikipedia: Inoue surface
---

## Example

Let $M$ be a matrix in the [[special linear group|0V9PMN0RBH]] $\mathrm{SL}(3, \Z)$ with one real [[eigenvalue|0BQAA2BK3Q]] $\alpha > 1$ and two non-real eigenvalues $\beta, \bar\beta$.
Let $a = (a_1, a_2, a_3)^T \in \R^3$ be an eigenvector of $M$ for $\alpha$ and $b = (b_1, b_2, b_3)^T \in \C^3$ an eigenvector for $\beta$.
Let $\mathbb{H}$ be the [[upper half-plane|X6XGGH5FPW]] and let $G_M$ be the group of holomorphic automorphisms of $\mathbb{H} \times \C$ generated by
$$g_0(w, z) = (\alpha w, \beta z), \qquad g_i(w, z) = (w + a_i, z + b_i) \quad (i = 1, 2, 3).$$
The quotient $S_M = (\mathbb{H} \times \C)/G_M$ is the *Inoue surface* of type $S_M$, a compact complex surface.

Check of the group structure: from $M a = \alpha a$ and $M b = \beta b$ one gets $\alpha a_i = \sum_j m_{ij} a_j$ and $\beta b_i = \sum_j m_{ij} b_j$.
Hence $g_0 g_i g_0^{-1} = g_1^{m_{i1}} g_2^{m_{i2}} g_3^{m_{i3}}$, so the translations form a normal subgroup isomorphic to $\Z^3$, and $G_M$ is a semidirect product $\Z^3 \rtimes \Z$.
The maps $g_i$ keep $\operatorname{Im} w$ fixed, because $a_i$ is real, and $g_0$ multiplies it by $\alpha$; so $G_M$ preserves $\mathbb{H} \times \C$.

A concrete $M$: the companion matrix $M = \begin{pmatrix} 0 & 0 & 1 \\ 1 & 0 & 1 \\ 0 & 1 & 0 \end{pmatrix}$ of $x^3 - x - 1$ has determinant $1$.
The discriminant of $x^3 - x - 1$ is $-4(-1)^3 - 27 = -23 < 0$, so it has one real root $\alpha \approx 1.3247 > 1$ and two non-real roots.

## Remarks

The map $(w, z) \mapsto \log \operatorname{Im} w$ descends to a map $S_M \to \R / (\log\alpha)\Z$, which makes $S_M$ a bundle over a circle with fibre a real $3$-torus.
Its Betti numbers are $b_1 = 1$ and $b_2 = 0$, it contains no compact complex curves, and its Kodaira dimension is $-\infty$, so it is a [[class VII surface|Y4NSSJT4RC]].
Its universal cover is $\mathbb{H} \times \C$, not $\C^2 \setminus \set{0}$, so it is not a [[Hopf surface|PNFNGBXRM3]].
Inoue also introduced two further families, $S^{+}$ and $S^{-}$, built in a similar way from integer matrices.
A theorem of Bogomolov states that a minimal class VII surface with $b_2 = 0$ is a Hopf surface or an Inoue surface.
Li, Yau and Zheng, and later Teleman, gave complete proofs.

## Explanations

### L1 human

An Inoue surface is built from the product of the upper half-plane and the complex line, divided by a group made from an integer $3 \times 3$ matrix.
The group shifts points in three independent directions and stretches by the eigenvalues of the matrix.
The result is a compact complex surface that contains no complex curves at all.
Together with the Hopf surfaces, these surfaces are the class VII surfaces whose second Betti number is zero.