Inoue introduced three families of surfaces,
S0,
S+ and
S−, which are compact quotients of \Complex \times \mathbb{H} (a product of a
complex plane by a half-plane). These Inoue surfaces are
solvmanifolds. They are obtained as quotients of \Complex \times \mathbb{H} by a solvable
discrete group which acts holomorphically on \Complex \times \mathbb{H}. The solvmanifold surfaces constructed by Inoue all have second
Betti number b_2=0. These surfaces are of
Kodaira class VII, which means that they have b_1=1 and
Kodaira dimension -\infty. It was proven by
Bogomolov, Li–
Yau and Teleman that any
surface of class VII with b_2=0 is a
Hopf surface or an Inoue-type solvmanifold. These surfaces have no meromorphic functions and no curves. K. Hasegawa gives a list of all complex 2-dimensional solvmanifolds; these are
complex torus,
hyperelliptic surface,
Kodaira surface and Inoue surfaces
S0,
S+ and
S−. The Inoue surfaces are constructed explicitly as follows.
Of type S0 Let
φ be an integer 3 × 3 matrix, with two complex eigenvalues \alpha, \overline{\alpha} and a real eigenvalue
c > 1, with |\alpha|^2c=1. Then
φ is invertible over integers, and defines an action of the group of integers, \Z, on \Z^3. Let \Gamma:=\Z^3\rtimes\Z. This group is a lattice in
solvable Lie group :\R^3\rtimes\R = (\C \times\R ) \rtimes\R, acting on \C \times \R, with the (\C \times\R )-part acting by translations and the \rtimes\R -part as (z,r) \mapsto (\alpha^tz, c^tr). We extend this action to \C \times \mathbb{H} = \C \times \R \times \R^{>0} by setting v \mapsto e^{\log ct} v, where
t is the parameter of the \rtimes\R-part of \R^3\rtimes\R, and acting trivially with the \R^3 factor on \R^{>0}. This action is clearly holomorphic, and the quotient \C \times \mathbb{H}/\Gamma is called
Inoue surface of type S^0. The Inoue surface of type
S0 is determined by the choice of an integer matrix
φ, constrained as above. There is a countable number of such surfaces.
Of type S+ Let
n be a positive integer, and \Lambda_n be the group of upper triangular matrices :\begin{bmatrix} 1 & x & z/n \\ 0 & 1 & y \\ 0 & 0 & 1 \end{bmatrix}, \qquad x,y,z \in \Z. The quotient of \Lambda_n by its center
C is \Z^2. Let
φ be an automorphism of \Lambda_n, we assume that
φ acts on \Lambda_n/C=\Z^2 as a matrix with two positive real eigenvalues
a, b, and
ab = 1. Consider the solvable group \Gamma_n := \Lambda_n\rtimes \Z, with \Z acting on \Lambda_n as
φ. Identifying the group of upper triangular matrices with \R^3, we obtain an action of \Gamma_n on \R^3= \C \times \R. Define an action of \Gamma_n on \C \times \mathbb{H}= \C \times \R \times \R^{>0} with \Lambda_n acting trivially on the \R^{>0}-part and the \Z acting as v \mapsto e^{t \log b}v. The same argument as for Inoue surfaces of type S^0 shows that this action is holomorphic. The quotient \C \times \mathbb{H}/\Gamma_n is called
Inoue surface of type S^+.
Of type S− Inoue surfaces of type S^- are defined in the same way as for
S+, but two eigenvalues
a, b of
φ acting on \Z^2 have opposite sign and satisfy
ab = −1. Since a square of such an endomorphism defines an Inoue surface of type
S+, an Inoue surface of type
S− has an unramified double cover of type
S+. ==Parabolic and hyperbolic Inoue surfaces==