On the classification of Inoue surfaces

TL;DR

This paper proves the uniqueness of holomorphic connections on Inoue surfaces and provides an explicit classification of these surfaces based on matrix conjugacy classes, resulting in a detailed biholomorphic classification.

Contribution

It establishes the uniqueness of holomorphic connections on Inoue surfaces and classifies them explicitly using matrix conjugacy classes, linking algebraic data to geometric types.

Findings

Unique holomorphic connection on any Inoue surface.

Classification of Inoue surfaces via matrix conjugacy classes.

Explicit parameterization of deformation and biholomorphism classes.

Abstract

We prove that any Inoue surface admits a unique holomorphic connection. Using this result we show that two Inoue surfaces $S=H\times\mathbb{C}/G$, $S'=H\times\mathbb{C}/G'$ are biholomorphic if and only if $G$, $G'$ are conjugate in the group of affine transformations of $H\times\mathbb{C}$. This result allows us to prove explicit classification theorems for Inoue surfaces: Let $\mathcal{M}$ be the set of ${\rm SL}(3,\mathbb{Z})$-matrices $M$ with a real eigenvalue $\alpha>1$ and two non-real eigenvalues, and $\mathcal{N}^\pm $ the set of ${\rm GL}(2,\mathbb{Z})$-matrices $N$ with a real eigenvalue $\alpha>1$ and $\det(N)=\pm 1$. We prove that: For any ${\rm GL}(3,\mathbb{Z})$-similarity class $\mathfrak{M}\in \mathcal{M}/\sim$, there exists exactly two biholomorphism classes of type I Inoue surfaces. For any ${\rm GL}(2,\mathbb{Z})$ similarity class $\mathfrak{N}=[N]\in \mathcal{N}^+/\sim$ and positive integer $r\in\mathbb{N}^*$, we have a finite set of deformation classes of type II Inoue surfaces. This set is parameterised by the quotient of $\mathbb{Z}^2/(I_2-N)\mathbb{Z}^2+r\mathbb{Z}^2$ by an action of the "positive centraliser" $Z^+_{{\rm GL}(2,\mathbb{Z})}(N)$ of $N$ in ${\rm GL}(2,\mathbb{Z})$. The set of biholomorphism types corresponding to a deformation class, endowed with its natural topology, can be identified with either $\mathbb{C}^*$ or $\mathbb{C}$. For any ${\rm GL}(2,\mathbb{Z})$-similarity class $\mathfrak{N}=[N]\in \mathcal{N}^-/\sim$ and positive integer $r\in\mathbb{N}^*$, we have a finite set of biholomorphism classes of type III Inoue surfaces. This set is parameterised by the quotient of $\mathbb{Z}^2/(I_2+N)\mathbb{Z}^2+r\mathbb{Z}^2$ by an action of $Z^+_{{\rm GL}(2,\mathbb{Z})}(N)$. In both cases the group $Z^+_{{\rm GL}(2,\mathbb{Z})}(N)$ is infinite cyclic (see section 5).