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 introduced three families of surfaces, S⁰, S⁺ and S⁻, which are compact quotients of ${\displaystyle \mathbb {C} \times \mathbb {H} }${\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 ${\displaystyle \mathbb {C} \times \mathbb {H} }${\displaystyle \mathbb {C} \times \mathbb {H} } by a solvable discrete group which acts holomorphically on ${\displaystyle \mathbb {C} \times \mathbb {H} .}${\displaystyle \mathbb {C} \times \mathbb {H} .}

The solvmanifold surfaces constructed by Inoue all have second Betti number ${\displaystyle b_{2}=0}${\displaystyle b_{2}=0}. These surfaces are of Kodaira class VII, which means that they have ${\displaystyle b_{1}=1}${\displaystyle b_{1}=1} and Kodaira dimension ${\displaystyle -\infty }${\displaystyle -\infty }. It was proven by Bogomolov,^[2] Li–Yau ^[3] and Teleman^[4] that any surface of class VII with ${\textstyle b_{2}=0}${\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]

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

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

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

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

Let n be a positive integer, and ${\displaystyle \Lambda _{n}}${\displaystyle \Lambda _{n}} be the group of upper triangular matrices

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

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

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

• Complex surfaces

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