Inoue surface

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 ${\mathbb {C} }\times H$ ${\ displaystyle {\ mathbb {C}} \ times H}$ (products of the complex plane on the half-plane). These Inoue surfaces are . They are obtained as a factor. ${\mathbb {C} }\times H$ ${\ displaystyle {\ mathbb {C}} \ times H}$ on a solvable discrete group that acts holomorphically on ${\mathbb {C} }\times H$ ${\ displaystyle {\ mathbb {C}} \ times H}$ .

All solvable surfaces that Inoue built have a second Betty number. $b_{2}=0$ ${\ displaystyle b_ {2} = 0}$ . These surfaces are , which means that for them $b_{1}=1$ ${\ displaystyle b_ {1} = 1}$ and the is equal to $-\infty$ ${\ displaystyle - \ infty}$ . 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 $\phi$ ${\ displaystyle \ phi}$ will be an integer 3 × 3 matrix with two complex eigenvalues $\alpha ,{\bar {\alpha }}$ ${\ displaystyle \ alpha, {\ bar {\ alpha}}}$ and real eigenvalue c> 1 , while $|\alpha |^{2}c=1$ ${\ displaystyle | \ alpha | ^ {2} c = 1}$ . Then $\phi$ ${\ displaystyle \ phi}$ reversible in integers and defines the action of the group ${\mathbb {Z} }$ ${\ displaystyle {\ mathbb {Z}}}$ integers on ${\mathbb {Z} }^{3}$ ${\ displaystyle {\ mathbb {Z}} ^ {3}}$ . Let be $\Gamma :={\mathbb {Z} }^{3}\rtimes {\mathbb {Z} }$ ${\ displaystyle \ Gamma: = {\ mathbb {Z}} ^ {3} \ rtimes {\ mathbb {Z}}}$ . This group is a lattice in a soluble Lie group.

{\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 ${\mathbb {C} }\times {\mathbb {R} }$ ${\ displaystyle {\ mathbb {C}} \ times {\ mathbb {R}}}$ while the group acts on $({\mathbb {C} }\times {\mathbb {R} })$ ${\ displaystyle ({\ mathbb {C}} \ times {\ mathbb {R}})}$ -part by hyphenation, and on $\rtimes {\mathbb {R} }$ ${\ displaystyle \ rtimes {\ mathbb {R}}}$ -part like $(z,r)\mapsto (\alpha ^{t}z,c^{t}r)$ ${\ displaystyle (z, r) \ mapsto (\ alpha ^ {t} z, c ^ {t} r)}$ .

We extend this action by ${\mathbb {C} }\times H={\mathbb {C} }\times {\mathbb {R} }\times {\mathbb {R} }^{>0}$ ${\ displaystyle {\ mathbb {C}} \ times H = {\ mathbb {C}} \ times {\ mathbb {R}} \ times {\ mathbb {R}} ^ {> 0}}$ by putting $v\mapsto e^{\log ct}v$ ${\ displaystyle v \ mapsto e ^ {\ log ct} v}$ where t is a parameter $\rtimes {\mathbb {R} }$ ${\ displaystyle \ rtimes {\ mathbb {R}}}$ - group parts ${\mathbb {R} }^{3}\rtimes {\mathbb {R} }$ ${\ displaystyle {\ mathbb {R}} ^ {3} \ rtimes {\ mathbb {R}}}$ . The action is trivial on the factor. ${\mathbb {R} }^{3}$ ${\ displaystyle {\ mathbb {R}} ^ {3}}$ by ${\mathbb {R} }^{>0}$ ${\ displaystyle {\ mathbb {R}} ^ {> 0}}$ . This action is obviously holomorphic and the factor ${\mathbb {C} }\times H/\Gamma$ ${\ displaystyle {\ mathbb {C}} \ times H / \ Gamma}$ called the Inoue surface of type S ⁰ .

The Inoue surface S ⁰ is determined by the choice of the integer matrix $\phi$ ${\ displaystyle \ phi}$ , with the above limitations. There is a countable number of such surfaces.

S ⁺ Surfaces

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

{\begin{bmatrix}1&x&{\frac {z}{n}}\\0&1&y\\0&0&1\end{bmatrix}},

{\ displaystyle {\ begin {bmatrix} 1 & x & {\ frac {z} {n}} \\ 0 & 1 & y \\ 0 & 0 & 1 \ end {bmatrix}},}

,

where x, y, z are integers. Consider an automorphism $\Lambda _{n}$ ${\ displaystyle \ Lambda _ {n}}$ which we denote $\phi$ ${\ displaystyle \ phi}$ . Group factor $\Lambda _{n}$ ${\ displaystyle \ Lambda _ {n}}$ in its center C is ${\mathbb {Z} }^{2}$ ${\ displaystyle {\ mathbb {Z}} ^ {2}}$ . Let's pretend that $\phi$ ${\ displaystyle \ phi}$ acts on $\Lambda _{n}/C={\mathbb {Z} }^{2}$ ${\ displaystyle \ Lambda _ {n} / C = {\ mathbb {Z}} ^ {2}}$ as a matrix with two positive real eigenvalues a, b , with ab = 1.

Consider a soluble group $\Gamma _{n}:=\Lambda _{n}\rtimes {\mathbb {Z} }$ ${\ displaystyle \ Gamma _ {n}: = \ Lambda _ {n} \ rtimes {\ mathbb {Z}}}$ , with ${\mathbb {Z} }$ ${\ displaystyle {\ mathbb {Z}}}$ acting on $\Lambda _{n}$ ${\ displaystyle \ Lambda _ {n}}$ , as $\phi$ ${\ displaystyle \ phi}$ . Identifying the group of upper triangular matrices with ${\mathbb {R} }^{3}$ ${\ displaystyle {\ mathbb {R}} ^ {3}}$ we will get action $\Gamma _{n}$ ${\ displaystyle \ Gamma _ {n}}$ on ${\mathbb {R} }^{3}={\mathbb {C} }\times {\mathbb {R} }$ ${\ displaystyle {\ mathbb {R}} ^ {3} = {\ mathbb {C}} \ times {\ mathbb {R}}}$ . Define action $\Gamma _{n}$ ${\ displaystyle \ Gamma _ {n}}$ on ${\mathbb {C} }\times H={\mathbb {C} }\times {\mathbb {R} }\times {\mathbb {R} }^{>0}$ ${\ displaystyle {\ mathbb {C}} \ times H = {\ mathbb {C}} \ times {\ mathbb {R}} \ times {\ mathbb {R}} ^ {> 0}}$ with $\Lambda _{n}$ ${\ displaystyle \ Lambda _ {n}}$ acting trivially on ${\mathbb {R} }^{>0}$ ${\ displaystyle {\ mathbb {R}} ^ {> 0}}$ -part and ${\mathbb {Z} }$ ${\ displaystyle {\ mathbb {Z}}}$ acts like $v\mapsto e^{t\log b}v$ ${\ displaystyle v \ mapsto e ^ {t \ log b} v}$ . The same arguments as for Inoue surfaces of type $S^{0}$ ${\ displaystyle S ^ {0}}$ , show that this action is holomorphic. Factor ${\mathbb {C} }\times H/\Gamma _{n}$ ${\ displaystyle {\ mathbb {C}} \ times H / \ Gamma _ {n}}$ called inoue surface type $S^{+}$ ${\ displaystyle S ^ {+}}$ .

Type S Surfaces ^-

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

[_5b5e82d269be2f20-1] Inoue, 1974 , p. 269-310.

[_7458d1c6f257c18a-2] Bogomolov, 1976 , p. 273-288.

[_bdd36270abe51a11-3] Li, Yau, 1987 , p. 560-573.

[_e0bc0e4dcd4403e7-4] Teleman, 1994 , p. 253-264.

[_c8170816df1f9cc0-5] ¹ ² Hasegawa, 2005 , p. 749-767.

[_5138d49b6151f658-6] Nakamura, 1984 , p. 393-443.

[_2681c5dc8d65ecb9-7] Nakamura, 2008 .

Inoue surface

Content

Inoue surfaces with b 2 = 0

Type S Surfaces 0

S + Surfaces

Type S Surfaces -