Hopf manifold

In complex geometry, Hopf manifold is obtainedas a quotient of the complex vector space(with zero deleted) $({Bbb C}^nackslash 0)$ by a free action of the group $Gamma cong {Bbb Z}$ of
integers, with the generator $gamma$ of $Gamma$ 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.

Examples

In a typical situation, $Gamma$ is generatedby a linear contraction, usually a diagonal matrix $qcdot Id$ , with $qin {Bbb C}$ a complex number, $0<|q|<1$ . Such manifoldis called "a classical Hopf manifold".

Properties

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

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 : $H=igg({Bbb C}^2ackslash 0igg)/Gamma,$ where $Gamma$ is a group generated bya polynomial contraction $gamma$ .Kodaira has found a normal form for $gamma$ .In appropriate coordinates, $gamma$ can be written as: $(x, y) mapsto (alpha x +lambda y^n, eta y)$ where $alpha, etain {Bbb C}$ are complex numberssatisfying $0<|alpha|leq |eta| <1$ , and either $;lambda=0$ or $;alpha^n=eta$ . 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 $S^3 imes S^1$ is biholomorphic to a Hopf surface.

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.

References

[1] K. Kodaira, "On the structure of compact complex analytic surfaces, II", American J. Math., 88 (1966), 682-722.

[2] K. Kodaira, [http://www.pnas.org/cgi/reprint/55/2/240.pdf Complex structures on $S^1 imes S^3$ ] , Proc. Nat. Acad. Sci. USA, 55 (1966), 240-243.

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