# Inoue surface

> Mediated Wiki article. Canonical URL: https://mediated.wiki/source/Inoue_surface
> Markdown URL: https://mediated.wiki/source/Inoue_surface.md
> Source: https://en.wikipedia.org/wiki/Inoue_surface
> Source revision: 1322530646
> License: Creative Commons Attribution-ShareAlike 4.0 International (https://creativecommons.org/licenses/by-sa/4.0/)

In [complex geometry](/source/complex_geometry), an '''Inoue surface''' is any of several [complex surface](/source/complex_surface)s of [Kodaira class VII](/source/Surfaces_of_class_VII). They are named after [Masahisa Inoue](/source/Masahisa_Inoue), who gave the first non-trivial examples of Kodaira class VII surfaces in 1974.<ref>M. Inoue, "On surfaces of class VII<sub>0</sub>," ''Inventiones math.'', 24 (1974), 269–310.</ref>

The Inoue surfaces are not [Kähler manifold](/source/K%C3%A4hler_manifold)s.

==Inoue surfaces with ''b''<sub>2</sub> = 0==
Inoue introduced three families of surfaces, ''S''<sup>0</sup>, ''S''<sup>+</sup> and ''S''<sup>−</sup>, which are compact quotients
of <math>\Complex \times \mathbb{H}</math> (a product of a [complex plane](/source/complex_plane) by a half-plane). These Inoue surfaces are [solvmanifold](/source/solvmanifold)s. They are obtained as quotients of <math>\Complex \times \mathbb{H}</math> by a solvable [discrete group](/source/discrete_group) which acts holomorphically on <math>\Complex \times \mathbb{H}.</math>

The solvmanifold surfaces constructed by Inoue all have second [Betti number](/source/Betti_number) <math>b_2=0</math>. These surfaces are of [Kodaira class VII](/source/Surfaces_of_class_VII), which means that they have <math>b_1=1</math> and [Kodaira dimension](/source/Kodaira_dimension) <math>-\infty</math>. It was proven by [Bogomolov](/source/Fedor_Bogomolov),<ref>Bogomolov, F.: "Classification of surfaces of class VII<sub>0</sub> with ''b''<sub>2</sub>&nbsp;=&nbsp;0", ''Math. USSR Izv'' 10, 255&ndash;269 (1976)</ref> Li–[Yau](/source/Shing-Tung_Yau)<ref>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&ndash;573, World Scientific Publishing (1987)</ref> and Teleman<ref>Teleman, A.: "Projectively flat surfaces and Bogomolov's theorem on class VII<sub>0</sub>-surfaces", ''Int. J. Math.'', Vol. 5, No 2, 253&ndash;264 (1994)</ref> that any [surface of class VII](/source/Surfaces_of_class_VII) with <math display="inline">b_2=0</math> is a [Hopf surface](/source/Hopf_surface) or an Inoue-type solvmanifold.

These surfaces have no meromorphic functions and no curves.

K. Hasegawa <ref name="hasegawa">Keizo Hasegawa [https://arxiv.org/abs/0804.4223 ''Complex and Kähler structures on Compact Solvmanifolds,''] J. Symplectic Geom. Volume 3, Number 4 (2005), 749–767.</ref> gives a list of all complex 2-dimensional solvmanifolds; these are [complex torus](/source/complex_torus), [hyperelliptic surface](/source/hyperelliptic_surface), [Kodaira surface](/source/Kodaira_surface) and Inoue surfaces ''S''<sup>0</sup>, ''S''<sup>+</sup> and ''S''<sup>−</sup>.

The Inoue surfaces are constructed explicitly as follows.<ref name="hasegawa" />

===Of type ''S''<sup>0</sup>===
Let ''φ'' be an integer 3&nbsp;×&nbsp;3 matrix, with two complex eigenvalues <math>\alpha, \overline{\alpha}</math> and a real eigenvalue ''c''&nbsp;>&nbsp;1, with <math>|\alpha|^2c=1</math>. Then ''φ'' is invertible over integers, and defines an action of the group of integers, <math>\Z,</math> on <math>\Z^3</math>. Let <math>\Gamma:=\Z^3\rtimes\Z.</math> This group is a lattice in [solvable](/source/solvable_group) [Lie group](/source/Lie_group)

:<math>\R^3\rtimes\R = (\C \times\R ) \rtimes\R,</math>

acting on <math>\C \times \R,</math> with the <math>(\C \times\R )</math>-part acting by translations and the <math>\rtimes\R </math>-part as <math>(z,r) \mapsto (\alpha^tz, c^tr).</math>

We extend this action to <math>\C \times \mathbb{H} = \C \times \R \times \R^{>0}</math> by setting <math>v \mapsto e^{\log ct} v</math>, where ''t'' is the parameter of the <math>\rtimes\R</math>-part of <math>\R^3\rtimes\R,</math> and acting trivially with the <math>\R^3</math> factor on <math>\R^{>0}</math>. This action is clearly holomorphic, and the quotient <math>\C \times \mathbb{H}/\Gamma</math> is called '''Inoue surface of type''' <math>S^0.</math>

The Inoue surface of type ''S''<sup>0</sup> is determined by the choice of an integer matrix ''φ'', constrained as above. There is a countable number of such surfaces.

===Of type ''S''<sup>+</sup>===
Let ''n'' be a positive integer, and <math>\Lambda_n</math> be the group of upper triangular matrices

:<math>\begin{bmatrix}
1 & x & z/n \\
0 & 1 & y \\
0 & 0 & 1 \end{bmatrix}, \qquad x,y,z \in \Z.</math>

The quotient of <math>\Lambda_n</math> by its center ''C'' is <math>\Z^2</math>. Let ''φ'' be an automorphism of <math>\Lambda_n</math>, we assume that ''φ'' acts on <math>\Lambda_n/C=\Z^2</math> as a matrix with two positive real eigenvalues ''a, b'', and ''ab''&nbsp;=&nbsp;1. Consider the solvable group <math>\Gamma_n := \Lambda_n\rtimes \Z,</math> with <math>\Z</math> acting on <math>\Lambda_n</math> as ''φ''. Identifying the group of upper triangular matrices with <math>\R^3,</math> we obtain an action of <math>\Gamma_n</math> on <math>\R^3= \C \times \R.</math> Define an action of <math>\Gamma_n</math> on <math>\C \times \mathbb{H}= \C \times \R \times \R^{>0}</math> with <math>\Lambda_n</math> acting trivially on the <math>\R^{>0}</math>-part and the <math>\Z</math> acting as <math>v \mapsto e^{t \log b}v.</math> The same argument as for Inoue surfaces of type <math>S^0</math> shows that this action is holomorphic. The quotient <math>\C \times \mathbb{H}/\Gamma_n</math> is called '''Inoue surface of type''' <math>S^+.</math>

===Of type ''S''<sup>−</sup>===
'''Inoue surfaces of type''' <math>S^-</math> are defined in the same way as for ''S''<sup>+</sup>, but two eigenvalues ''a, b'' of ''φ'' acting on <math>\Z^2</math> have opposite sign and satisfy ''ab''&nbsp;=&nbsp;−1. Since a square of such an endomorphism defines an Inoue surface of type ''S''<sup>+</sup>, an Inoue surface of type ''S''<sup>−</sup> has an unramified double cover of type ''S''<sup>+</sup>.

==Parabolic and hyperbolic Inoue surfaces==
Parabolic and hyperbolic Inoue surfaces are Kodaira class VII surfaces defined by [Iku Nakamura](/source/Iku_Nakamura) in 1984.<ref>I. Nakamura, "On surfaces of class VII<sub>0</sub> with curves," ''Inv. Math.'' 78, 393&ndash;443 (1984).</ref> They are not solvmanifolds. These surfaces have positive second Betti number. They have [spherical shell](/source/Spherical_shell_conjecture)s, and can be deformed into a blown-up [Hopf surface](/source/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 VII<sub>0</sub> surfaces with two cycles of rational curves.<ref>I. Nakamura. "[http://www.math.sci.hokudai.ac.jp/~nakamura/70surfaces080306.pdf Survey on VII<sub>0</sub> surfaces] {{Webarchive|url=https://web.archive.org/web/20110716192501/http://www.math.sci.hokudai.ac.jp/~nakamura/70surfaces080306.pdf |date=16 July 2011 }}", ''Recent Developments in NonKaehler Geometry'', Sapporo, 2008 March.</ref> 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.<ref>G. Dloussky, "Une construction elementaire des surfaces d'Inoue–Hirzebruch". ''Math. Ann.'' 280, 663–682 (1988).</ref>

==Notes==
{{Reflist}}
{{Authority control}}

{{Use dmy dates|date=April 2020}}

{{DEFAULTSORT:Inoue Surface}}
Category:Complex surfaces

---
Adapted from the Wikipedia article [Inoue surface](https://en.wikipedia.org/wiki/Inoue_surface) by Wikipedia contributors ([contributor history](https://en.wikipedia.org/wiki/Inoue_surface?action=history)). Available under [Creative Commons Attribution-ShareAlike 4.0 International](https://creativecommons.org/licenses/by-sa/4.0/). Changes may have been made.