Inoue surface

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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 {{\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 {{\mathbb C}}\times H by a solvable discrete group which acts holomorphically on {{\mathbb C}}\times H.

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

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

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

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

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

Consider the solvable group \Gamma _{n}:=\Lambda _{n}\ltimes {{\mathbb Z}}, with {{\mathbb Z}} acting on \Lambda _{n} as φ. Identifying the group of upper triangular matrices with {{\mathbb R}}^{3}, we obtain an action of \Gamma _{n} on {{\mathbb R}}^{3}={{\mathbb C}}\times {{\mathbb R}}. Define an action of \Gamma _{n} on {{\mathbb C}}\times H={{\mathbb C}}\times {{\mathbb R}}\times {{\mathbb R}}^{{>0}} with \Lambda _{n} acting trivially on the {{\mathbb R}}^{{>0}}-part and the {{\mathbb Z}} acting as 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 {{\mathbb C}}\times H/\Gamma _{n} is called Inoue surface of type S^{+}.

Inoue surfaces of type S−

Inoue surfaces of type S^{-} are defined in the same was as for S+, but two eigenvalues a, b of φ acting on {{\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 are also known as half-Inoue surfaces. These surfaces can be defined as class VII0 (that is, class VII and minimal) surfaces with an elliptic curve and a cycle 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. ↑ 5.0 5.1 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.
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