Hopf manifold

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In complex geometry, a Hopf manifold (Hopf 1948) is obtained as a quotient of the complex vector space (with zero deleted) $({{\mathbb C}}^{n}\backslash 0)$ by a free action of the group $\Gamma \cong {{\mathbb Z}}$ of integers, with the generator $\gamma$ of $\Gamma$ acting by holomorphic contractions. Here, a holomorphic contraction is a map $\gamma :\;{{\mathbb C}}^{n}\mapsto {{\mathbb C}}^{n}$ such that a sufficiently big iteration $\;\gamma ^{N}$ puts any given compact subset ${{\mathbb C}}^{n}$ onto an arbitrarily small neighbourhood of 0.

Two dimensional Hopf manifolds are called Hopf surfaces.

Examples

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

Properties

A Hopf manifold $H:=({{\mathbb C}}^{n}\backslash 0)/{{\mathbb Z}}$ is diffeomorphic to $S^{{2n-1}}\times S^{1}$ . For $n\geq 2$ , 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.

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

Hopf, Heinz (1948), "Zur Topologie der komplexen Mannigfaltigkeiten", Studies and Essays Presented to R. Courant on his 60th Birthday, January 8, 1948, Interscience Publishers, Inc., New York, pp. 167–185, MR 0023054
Ornea, L. (2001), "H/h110270", in Hazewinkel, Michiel, Encyclopedia of Mathematics, Springer, ISBN 978-1-55608-010-4

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