On the Classification of Inoue Surfaces
Vol. 33 (2026), No. 3, Page 283–357.
Khaled, Z. ; Teleman, A.
On the Classification of Inoue Surfaces
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Abstract:
We prove that any Inoue surface admits a unique holomorphic connection and we infer 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 $\mathrm{SL}(3,\mathbb{Z})$-matrices $M$ with a real eigenvalue $\alpha>1$ and two non-real eigenvalues, and let $\mathcal{N}^\pm $ the set of $\mathrm{GL}(2,\mathbb{Z})$-matrices $N$ with a real eigenvalue $\alpha>1$ 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 $r\in\mathbb{N}^*$, 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 $\mathbb{C}^*$ or $\mathbb{C}$.
- (-) For any $\mathrm{GL}(2,\mathbb{Z})$-similarity class $\mathfrak{N}=[N]\in \mathcal{N}^-/\sim$ and positive integer $r\in\mathbb{N}^*$, 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 $Z^+_{\mathrm{GL}(2,\mathbb{Z})}(N)$-can be identified with the stabiliser of $N$ in $\mathrm{PGL}(2,\mathbb{Z})$ and is an infinite cyclic group (see section 5).
Taking into account the Latimer-MacDuffee theorem and a classical finiteness theorem for ideal classes in orders, it follows that:
- For any polynomial $\chi\in \chi(\mathcal{M})$ we have only finitely many biholomorphism classes of type I surfaces.
- For any pair $(\chi,r)\in \chi(\mathcal{N}^+)\times\mathbb{N}^*$ we have only finitely many deformation classes of type II Inoue surfaces.
- For any pair $(\chi,r)\in \chi(\mathcal{N}^-)\times\mathbb{N}^*$ we have only finitely many biholomorphism classes of type III Inoue surfaces.
Keywords: Inoue surfaces, complex surfaces.
Mathematics Subject Classification (2020): 32J15, 32Q57.
Received: 2024-07-22
