- Hopf manifold
In
complex geometry , Hopf manifold is obtainedas a quotient of the complexvector space (with zero deleted) by a free action of the group ofinteger s, with the generator of acting by holomorphic contractions. Here, a "holomorphic contraction"is a map such that a sufficiently big iteration puts any given compact subset onto an arbitrarily small neighbourhood of 0.Examples
In a typical situation, is generatedby a linear contraction, usually a diagonal matrix , with a complex number, . Such manifoldis called "a classical Hopf manifold".
Properties
A Hopf manifold is diffeomorphic to .It is non-Kähler. Indeed, the first cohomology group of "H"is odd-dimensional. By
Hodge decomposition ,odd cohomology of a compactKähler manifold are 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 : where is a group generated bya polynomial contraction .Kodaira has found a normal form for .In appropriate coordinates, can be written as:where are complex numberssatisfying , and either or . When , 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 is biholomorphic to a Hopf surface.
Hypercomplex structure
Even-dimensional Hopf manifolds admit
hypercomplex structure.The Hopf surface is the only compacthypercomplex 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 ] , Proc. Nat. Acad. Sci. USA, 55 (1966), 240-243.
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