Inoue surface explained

In complex geometry, an Inoue surface is any of 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 quotientsof

\Complex x H (a product of a complex plane by a half-plane). These Inoue surfaces are solvmanifolds. They are obtained as quotients of \Complex x H by a solvable discrete group which acts holomorphically on \Complex x H . b _{2=0} . These surfaces are of Kodaira class VII, which means that they have b _{1=1} and Kodaira dimension -inf ty . It was proven by Bogomolov,^[2] Li–Yau^[3] and Teleman^[4] 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 ^[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.

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 x \R ) \rtimes\R,

acting on

\C x \R, with the (\C x \R ) -part acting by translations and the \rtimes\R -part as (z,r) \mapsto (\alpha ^{tz,} c ^{tr).}

We extend this action to

\C x H = \C x \R x \R ^{>0} by setting v \mapsto e ^{log} 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 x 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

Λ _{n} be the group of upper triangular matrices\begin{bmatrix} 1 & x & z/n \\ 0 & 1 & y \\ 0 & 0 & 1 \end{bmatrix}, x,y,z \in \Z.

The quotient of

Λ _{n} by its center C is \Z ^{2} . Let φ be an automorphism of Λ _{n} , we assume that φ acts on Λ ^{2} _{n/C = \Z} as a matrix with two positive real eigenvalues a, b, and ab = 1. Consider the solvable group \Gamma _{n} : = Λ _{n\rtimes} \Z, with \Z acting on Λ _{n} as φ. Identifying the group of upper triangular matrices with \R ^{3,} we obtain an action of \Gamma _{n} on \R ^{3=} \C x \R. Define an action of \Gamma _{n} on \C x H = \C x \R x \R ^{>0} with Λ _{n} acting trivially on the \R ^{>0} -part and the \Z acting as v \mapsto e ^{t} v. The same argument as for Inoue surfaces of type S ^{0} shows that this action is holomorphic. The quotient \C x 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

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] Parabolic and hyperbolic surfaces are particular cases of minimal surfaces with global spherical shells (GSS) also called Kato surfaces. All these surfaces may be constructed by non invertible contractions.^[8]

Notes and References

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-Kähler 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 Kähler 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.
8. G. Dloussky, "Une construction elementaire des surfaces d'Inoue–Hirzebruch". Math. Ann. 280, 663–682 (1988).