Inoue surface

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

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 ${\displaystyle -\infty }$. It was proven by Bogomolov,[2] Li–Yau [3] and Teleman[4] that any surface of class VII with ${\textstyle b_{2}=0}$ 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.[5]

Of type S⁰

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

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

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

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

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 ${\displaystyle \Lambda _{n}}$ be the group of upper triangular matrices

${\displaystyle {\begin{bmatrix}1&x&z/n\\0&1&y\\0&0&1\end{bmatrix}},\qquad x,y,z\in \mathbb {Z} .}$

The quotient of ${\displaystyle \Lambda _{n}}$ by its center C is ${\displaystyle \mathbb {Z} ^{2}}$. Let φ be an automorphism of ${\displaystyle \Lambda _{n}}$, 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 ${\displaystyle \Gamma _{n}:=\Lambda _{n}\rtimes \mathbb {Z} ,}$ with ${\displaystyle \mathbb {Z} }$ acting on ${\displaystyle \Lambda _{n}}$ as φ. Identifying the group of upper triangular matrices with ${\displaystyle \mathbb {R} ^{3},}$ we obtain an action of ${\displaystyle \Gamma _{n}}$ on ${\displaystyle \mathbb {R} ^{3}=\mathbb {C} \times \mathbb {R} .}$ Define an action of ${\displaystyle \Gamma _{n}}$ on ${\displaystyle \mathbb {C} \times \mathbb {H} =\mathbb {C} \times \mathbb {R} \times \mathbb {R} ^{>0}}$ with ${\displaystyle \Lambda _{n}}$ acting trivially on the ${\displaystyle \mathbb {R} ^{>0}}$-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 ${\displaystyle S^{0}}$ shows that this action is holomorphic. The quotient ${\displaystyle \mathbb {C} \times \mathbb {H} /\Gamma _{n}}$ is called Inoue surface of type ${\displaystyle S^{+}.}$

Of type S⁻

Inoue surfaces of type ${\displaystyle S^{-}}$ are defined in the same way 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] 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

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).