On the classification of Inoue surfaces

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).