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.
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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
The solvmanifold surfaces constructed by Inoue all have second Betti number
These surfaces have no meromorphic functions and no curves.
K. Hasegawa 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
acting on
We extend this action to
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
where x, y, z are integers. Consider an automorphism of
Consider the solvable group
Of type S−
Inoue surfaces of type
Parabolic and hyperbolic Inoue surfaces
Parabolic and hyperbolic Inoue surfaces are Kodaira class VII surfaces defined by Iku Nakamura in 1984. 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.