Inoue surface

In complex geometry, a part of mathematics, the term Inoue surface denotes several complex surfaces of Kodaira class VII. They are named after Masahisa Inoue, who gave the first non-trivial examples of Kodaira class VII surfaces in 1974.^[1]

The Inoue surfaces are not Kähler manifolds.

Inoue surfaces with b₂ = 0

Inoue introduced three families of surfaces, S⁰, S⁺ and S⁻, which are compact quotients of ${\Bbb C} \times H$ (a product of a complex plane by a half-plane). These Inoue surfaces are solvmanifolds. They are obtained as quotients of ${\Bbb C} \times H$ by a solvable discrete group which acts holomorphically on ${\Bbb C} \times H$.

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,^[2] Li-Yau ^[3] and Teleman^[4] that any surface of class VII with b₂ = 0 is a Hopf surface or an Inoue-type solvmanifold.

These surfaces have no meromorphic functions and no curves.

K. Hasegawa ^[5] 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.^[5]

Inoue surfaces of type S⁰

Let φ be an integer 3 × 3 matrix, with two complex eigenvalues $\alpha, \bar\alpha$ and a real eigenvalue c, with $|\alpha|^2c=1$. Then φ is invertible over integers, and defines an action of the group ${\Bbb Z}$ of integers on ${\Bbb Z}^3$. Let $\Gamma:={\Bbb Z}^3\ltimes{\Bbb Z}$. This group is a lattice in solvable Lie group

${\Bbb R}^3\ltimes{\Bbb R}= ({\Bbb C}\times{\Bbb R}) \ltimes{\Bbb R}$,

acting on ${\Bbb C} \times {\Bbb R}$, with the $({\Bbb C}\times{\Bbb R})$-part acting by translations and the $\ltimes{\Bbb R}$-part as $(z, r) \mapsto (\alpha^tz, c^tr)$.

We extend this action to ${\Bbb C} \times H= {\Bbb C} \times {\Bbb R} \times {\Bbb R}^{>0}$ by setting $v \mapsto e^{\log c t}v$, where t is the parameter of the $\ltimes{\Bbb R}$-part of ${\Bbb R}^3\ltimes{\Bbb R}$, and acting trivially with the ${\Bbb R}^3$ factor on ${\Bbb R}^{>0}$. This action is clearly holomorphic, and the quotient ${\Bbb C} \times H/\Gamma$ is called Inoue surface of type S⁰.

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.

Inoue surfaces of type S⁺

Let n be a positive integer, and $\Lambda_n$ be the group of upper triangular matrices

$\begin{bmatrix} 1 & x & \frac{z}{n} \\ 0 & 1 & y \\ 0 & 0 & 1 \end{bmatrix},$

where x, y, z are integers. Consider an automorphism of $\Lambda_n$, denoted as φ. The quotient of $\Lambda_n$ by its center C is ${\Bbb Z}^2$. We assume that φ acts on $\Lambda_n/C={\Bbb Z}^2$ as a matrix with two positive real eigenvalues a, b, and ab = 1.

Consider the solvable group $\Gamma_n := \Lambda_n\ltimes {\Bbb Z}$, with ${\Bbb Z}$ acting on $\Lambda_n$ as φ. Identifying the group of upper triangular matrices with ${\Bbb R}^3$, we obtain an action of $\Gamma_n$ on ${\Bbb R}^3= {\Bbb C}\times {\Bbb R}$. Define an action of $\Gamma_n$ on ${\Bbb C} \times H= {\Bbb C} \times {\Bbb R} \times {\Bbb R}^{>0}$ with $\Lambda_n$ acting trivially on the ${\Bbb R}^{>0}$-part and the ${\Bbb Z}$ 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 ${\Bbb C} \times H/\Gamma_n$ is called Inoue surface of type $S^+$.

Inoue surfaces of type S⁻

Inoue surfaces of type $S^-$ are defined in the same was as for S⁺, but two eigenvalues a, b of φ acting on ${\Bbb Z}^2$ 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

Parabolic and hyperbolic Inoue surfaces are Kodaira class VII surfaces defined by Iku Nakamura in 1984.^[6] They are not solvmanifolds. These surfaces have positive second Betti number. They have spherical shells, and can be deformed into a blown-up Hopf surface.

Parabolic Inoue surfaces are also known as half-Inoue surfaces. These surfaces can be defined as class VII₀ (that is, class VII and minimal) surfaces with an elliptic curve and a cycle of rational curves.

Hyperbolic Inoue surfaces are class VII₀ surfaces with two cycles of rational curves.^[7]

Notes

1. M. Inoue, On surfaces of class VII₀, Inventiones math., 24 (1974), 269–310.
2. Bogomolov, F.: Classification of surfaces of class VII₀ with b₂ = 0, Math. USSR Izv 10, 255–269 (1976)
3. Li, J., Yau, S., T.: Hermitian Yang-Mills connections on non-Kahler manifolds, Math. aspects of string theory (San Diego, Calif., 1986), Adv. Ser. Math. Phys. 1, 560–573, World Scientific Publishing (1987)
4. Teleman, A.: Projectively flat surfaces and Bogomolov's theorem on class VII₀-surfaces, Int. J. Math., Vol. 5, No 2, 253–264 (1994)
5. Keizo Hasegawa Complex and Kahler structures on Compact Solvmanifolds, J. Symplectic Geom. Volume 3, Number 4 (2005), 749–767.
6. I. Nakamura, On surfaces of class VII₀ with curves, Inv. Math. 78, 393–443 (1984).
7. I. Nakamura: Survey on VII₀ surfaces, Recent Developments in NonKaehler Geometry, Sapporo, 2008 March.