Hopf manifold

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In complex geometry, Hopf manifold is obtained as a quotient of the complex vector space (with zero deleted) $({\Bbb C}^n\backslash 0)$ by a free action of the group $\Gamma \cong {\Bbb Z}$ of integers, with the generator $γ$ of $Γ$ acting by holomorphic contractions. Here, a holomorphic contraction is a map $\gamma:\; {\Bbb C}^n \mapsto {\Bbb C}^n$ such that a sufficiently big iteration $\;\gamma^N$ puts any given compact subset ${\Bbb C}^n$ onto an arbitrarily small neighbourhood of 0.

1 Examples
2 Properties
3 Hopf surfaces
4 Hypercomplex structure
5 References

[edit] Examples

In a typical situation, $Γ$ is generated by a linear contraction, usually a diagonal matrix $q\cdot Id$ , with $q\in {\Bbb C}$ a complex number, $0 < | q | < 1$ . Such manifold is called a classical Hopf manifold.

[edit] Properties

A Hopf manifold $H:=({\Bbb C}^n\backslash 0)/{\Bbb Z}$ is diffeomorphic to $S^{2n-1}\times S^1$ . It is non-Kähler. Indeed, the first cohomology group of H is odd-dimensional. By Hodge decomposition, odd cohomology of a compact Kähler manifold are always even-dimensional.

[edit] Hopf surfaces

A 2-dimensional Hopf manifold is called a Hopf surface. In the course of classification of compact complex surfaces, Kodaira classified the Hopf surfaces, by splitting them into two subclasses, called class 0 Hopf surface and class 1 Hopf surfaces. A Hopf surface is obtained as

$H=\bigg({\Bbb C}^2\backslash 0\bigg)/\Gamma,$

where $Γ$ is a group generated by a polynomial contraction $γ$ . Kodaira has found a normal form for $γ$ . In appropriate coordinates, $γ$ can be written as

$(x, y) \mapsto (\alpha x +\lambda y^n, \beta y)$

where $\alpha, \beta\in {\Bbb C}$ are complex numbers satisfying $0<|\alpha|\leq |\beta| <1$ , and either $\;\lambda=0$ or $\;\alpha^n=\beta$ . When $\;\lambda=0$ , $H$ is called the Hopf surface of Kodaira class 1, otherwise - the Hopf surface of Kodaira class 0.

Kodaira has proven that any complex surface which is diffeomorphic to $S^3\times S^1$ is biholomorphic to a Hopf surface.

[edit] Hypercomplex structure

Even-dimensional Hopf manifolds admit hypercomplex structure. The Hopf surface is the only compact hypercomplex manifold of quaternionic dimension 1 which is not hyperkähler.