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 ${\displaystyle {\mathbb {C}}\times H}$ (a product of a complex plane by a half-plane). These Inoue surfaces are solvmanifolds. They are obtained as quotients of ${\displaystyle {\mathbb {C}}\times H}$ by a solvable discrete group which acts holomorphically on ${\displaystyle {\mathbb {C}}\times H}$.

The solvmanifold surfaces constructed by Inoue all have second Betti number ${\displaystyle b_{2}=0}$. These surfaces are of Kodaira class VII, which means that they have ${\displaystyle b_{1}=1}$ 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 ${\displaystyle \alpha ,{\bar {\alpha }}}$ and a real eigenvalue c, with ${\displaystyle |\alpha |^{2}c=1}$. Then φ is invertible over integers, and defines an action of the group ${\displaystyle {\mathbb {Z}}}$ of integers on ${\displaystyle {\mathbb {Z}}^{3}}$. Let ${\displaystyle \Gamma :={\mathbb {Z}}^{3}\ltimes {\mathbb {Z}}}$. This group is a lattice in solvable Lie group

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

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

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

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

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 ${\displaystyle {\mathbb {Z}}^{2}}$. We assume that φ acts on ${\displaystyle \Lambda _{n}/C={\mathbb {Z}}^{2}}$ as a matrix with two positive real eigenvalues a, b, and ab = 1.

Consider the solvable group ${\displaystyle \Gamma _{n}:=\Lambda _{n}\ltimes {\mathbb {Z}}}$, with ${\displaystyle {\mathbb {Z}}}$ acting on $\Lambda_n$ as φ. Identifying the group of upper triangular matrices with ${\displaystyle {\mathbb {R}}^{3}}$, we obtain an action of $\Gamma_n$ on ${\displaystyle {\mathbb {R}}^{3}={\mathbb {C}}\times {\mathbb {R}}}$. Define an action of $\Gamma_n$ on ${\displaystyle {\mathbb {C}}\times H={\mathbb {C}}\times {\mathbb {R}}\times {\mathbb {R}}^{>0}}$ with $\Lambda_n$ acting trivially on the ${\Bbb R}^{>0}$-part and the ${\displaystyle {\mathbb {Z}}}$ acting as ${\displaystyle v\mapsto e^{t\log b}v}$. The same argument as for Inoue surfaces of type $S^{0}$ shows that this action is holomorphic. The quotient ${\displaystyle {\mathbb {C}}\times H/\Gamma _{n}}$ is called Inoue surface of type $S^+$.

Inoue surfaces of type S⁻

Inoue surfaces of type ${\displaystyle S^{-}}$ are defined in the same was as for S⁺, but two eigenvalues a, b of φ acting on ${\displaystyle {\mathbb {Z}}^{2}}$ 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 contain a cycle of rational curves with 0 self- intersection and an elliptic curve. They are a particular case of Enoki surfaces which have a cycle of rational curves with zero self- intersection but without elliptic curve.

Half-Inoue surfaces contain a cycle C of rational curves and are a quotient of a hyperbolic Inoue surface with two cycles 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.