Hopf manifold
In complex geometry, a Hopf manifold (Hopf 1948) is obtained as a quotient of the complex vector space (with zero deleted) [math]\displaystyle{ ({\mathbb C}^n\backslash 0) }[/math] by a free action of the group [math]\displaystyle{ \Gamma \cong {\mathbb Z} }[/math] of integers, with the generator [math]\displaystyle{ \gamma }[/math] of [math]\displaystyle{ \Gamma }[/math] acting by holomorphic contractions. Here, a holomorphic contraction is a map [math]\displaystyle{ \gamma:\; {\mathbb C}^n \to {\mathbb C}^n }[/math] such that a sufficiently big iteration [math]\displaystyle{ \;\gamma^N }[/math] maps any given compact subset of [math]\displaystyle{ {\mathbb C}^n }[/math] onto an arbitrarily small neighbourhood of 0.
Two-dimensional Hopf manifolds are called Hopf surfaces.
Examples
In a typical situation, [math]\displaystyle{ \Gamma }[/math] is generated by a linear contraction, usually a diagonal matrix [math]\displaystyle{ q\cdot Id }[/math], with [math]\displaystyle{ q\in {\mathbb C} }[/math] a complex number, [math]\displaystyle{ 0\lt |q|\lt 1 }[/math]. Such manifold is called a classical Hopf manifold.
Properties
A Hopf manifold [math]\displaystyle{ H:=({\mathbb C}^n\backslash 0)/{\mathbb Z} }[/math] is diffeomorphic to [math]\displaystyle{ S^{2n-1}\times S^1 }[/math]. For [math]\displaystyle{ n\geq 2 }[/math], it is non-Kähler. In fact, it is not even symplectic because the second cohomology group is zero.
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
- Hazewinkel, Michiel, ed. (2001), "Hopf manifold", Encyclopedia of Mathematics, Springer Science+Business Media B.V. / Kluwer Academic Publishers, ISBN 978-1-55608-010-4, https://www.encyclopediaofmath.org/index.php?title=Main_Page
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