On the classification of Inoue surfaces

Abstract

We prove that any Inoue surface admits a unique holomorphic connection. Using this result we show that two Inoue surfaces $S=H\times \mathbb{C}/G$, $S^{\prime }=H\times \mathbb{C}/G^{\prime }$ are biholomorphic if and only if $G$, $G^{\prime }$ 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 $\mathrm{SL}(3,\mathbb{Z})$-matrices $M$ with a real eigenvalue $\hat{I}\pm >1$ and two non-real eigenvalues, and ${\mathcal{N}}^{\pm }$ the set of $\mathrm{GL}(2,\mathbb{Z})$-matrices $N$ with a real eigenvalue $\hat{I}\pm >1$ and $\det (N)=\pm 1$. We prove that: For any $\mathrm{GL}(3,\mathbb{Z})$-similarity class $\mathfrak{M}\in \mathcal{M}/\sim$, there exists exactly two biholomorphism classes of type I Inoue surfaces. For any $\mathrm{GL}(2,\mathbb{Z})$ similarity class $\mathfrak{N}=[N]\in {\mathcal{N}}^{+}/\sim$ and positive integer $r\in {\mathbb{N}}^{\ast }$, 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" $Z_{\mathrm{GL}(2,\mathbb{Z})}^{+}(N)$ of $N$ in $\mathrm{GL}(2,\mathbb{Z})$. The set of biholomorphism types corresponding to a deformation class, endowed with its natural topology, can be identified with either ${\mathbb{C}}^{\ast }$ or $\mathbb{C}$. For any $\mathrm{GL}(2,\mathbb{Z})$-similarity class $\mathfrak{N}=[N]\in {\mathcal{N}}^{-}/\sim$ and positive integer $r\in {\mathbb{N}}^{\ast }$, 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 $Z_{\mathrm{GL}(2,\mathbb{Z})}^{+}(N)$. In both cases the group $Z_{\mathrm{GL}(2,\mathbb{Z})}^{+}(N)$ is infinite cyclic (see section 5).

LaTeX, 43 pages. Revised version: We added relevant references in the introduction, and we inserted a new remark about the non-existence of Real structures on type I Inoue surfaces. New revision: minor corrections. To appear in the Journal of Mathematical Sciences, The University of Tokyo

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