The Inoue surface is some complex surface . The surfaces are named after Masahit Inoue, who cited the first non-trivial examples of Kodaira class VII surfaces in 1974 [1] .
Inoue surfaces are not Kähler varieties .
Content
Inoue surfaces with b 2 = 0
Inoue brought three families of surfaces, S 0 , S + and S - , which are compact factors (products of the complex plane on the half-plane). These Inoue surfaces are . They are obtained as a factor. on a solvable discrete group that acts holomorphically on .
All solvable surfaces that Inoue built have a second Betty number. . These surfaces are , which means that for them and the is equal to . As Bogomolov [2] , Li- Yau [3] and Telemann [4] proved, any with b 2 = 0 is a Hopf surface or a solvable variety of inoue type.
These surfaces have no meromorphic functions, as well as curves.
K. Hasegawa [5] gave a list of all complex two-dimensional soluble manifolds. These are the complex torus , the hyperelliptic surface , and the Inoue surface S 0 , S + and S - .
Inoue surfaces are constructed explicitly, as described below [5] .
Type S Surfaces 0
Let be will be an integer 3 × 3 matrix with two complex eigenvalues and real eigenvalue c> 1 , while . Then reversible in integers and defines the action of the group integers on . Let be . This group is a lattice in a soluble Lie group.
- ,
acting on while the group acts on -part by hyphenation, and on -part like .
We extend this action by by putting where t is a parameter - group parts . The action is trivial on the factor. by . This action is obviously holomorphic and the factor called the Inoue surface of type S 0 .
The Inoue surface S 0 is determined by the choice of the integer matrix , with the above limitations. There is a countable number of such surfaces.
S + Surfaces
Let n be a positive integer, and - group of upper triangular matrices
- ,
where x, y, z are integers. Consider an automorphism which we denote . Group factor in its center C is . Let's pretend that acts on as a matrix with two positive real eigenvalues a, b , with ab = 1.
Consider a soluble group , with acting on , as . Identifying the group of upper triangular matrices with we will get action on . Define action on with acting trivially on -part and acts like . The same arguments as for Inoue surfaces of type , show that this action is holomorphic. Factor called inoue surface type .
Type S Surfaces -
Inoue type surfaces are defined in the same way as S + , but the two eigenvalues a, b of the automorphism acting on , have opposite signs and the equality ab = −1 holds. Since the square of such an endomorphism defines an Inoue surface of type S + , the Inoue surface of type S - has an unbranched double covering of type S + .
Parabolic and hyperbolic surfaces of Inoue
The Inoue parabolic and hyperbolic surfaces are Kodaira class VII surfaces that Iku Nakamura defined in 1984 [6] . They are not solvable varieties. These surfaces have a positive second Betty number. The surfaces have spherical shells and can be deformed into a bulging Hopf surface .
Inoue parabolic surfaces contain a cycle of rational curves with 0 self-intersections and an elliptic curve. They are a special case of Enoki surfaces that have a cycle of rational curves with zero self-intersections but no elliptic curve. The Inoue semisurface contains a cycle C of rational curves and is a factor of the hyperbolic Inoue surface with two cycles of rational curves.
Inoue hyperbolic surfaces are class VII 0 surfaces with two cycles of rational curves [7] .
Notes
- ↑ Inoue, 1974 , p. 269-310.
- ↑ Bogomolov, 1976 , p. 273-288.
- ↑ Li, Yau, 1987 , p. 560-573.
- ↑ Teleman, 1994 , p. 253-264.
- ↑ 1 2 Hasegawa, 2005 , p. 749-767.
- ↑ Nakamura, 1984 , p. 393-443.
- ↑ Nakamura, 2008 .
Literature
- Keizo Hasegawa. Complex and Kahler structures on Compact Solvmanifolds // J. Symplectic Geom .. - 2005. - Vol. 3 , no. 4 - p . 749-767 .
- Bogomolov F. A. Classification of class surfaces of class VII 0 with b 2 = 0 // Izv. Academy of Sciences of the USSR. - 1976. - V. 40 , no. 2
- 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 .. - World Scientific Publishing, 1987. - T. 1.
- Nakamura I. On surfaces of class VII 0 with curves // Inventiones math .. - 1984. - T. 78 .
- Inoue M. On surfaces of class VII 0 // Inventiones math .. - 1974. - T. 24 .
- Teleman A. Projectively flat surfaces and Bogomolov's Theorem on Class VII 0 -surfaces // Int. J. Math .. - 1994. - V. 5 , no. 2
- Nakamura I. Survey on VII 0 surfaces // Recent Developments in NonKaehler Geometry . - Sapporo, 2008.