Hopf manifold: Difference between revisions
Salix alba (talk | contribs) →References: fix ref to SpringerEOM |
No edit summary |
||
Line 5: | Line 5: | ||
[[integer]]s, with the generator <math>\gamma</math> |
[[integer]]s, with the generator <math>\gamma</math> |
||
of <math>\Gamma</math> acting by holomorphic [[Contraction mapping|contractions]]. Here, a ''holomorphic contraction'' |
of <math>\Gamma</math> acting by holomorphic [[Contraction mapping|contractions]]. Here, a ''holomorphic contraction'' |
||
is a map <math>\gamma:\; {\mathbb C}^n \ |
is a map <math>\gamma:\; {\mathbb C}^n \to {\mathbb C}^n</math> |
||
such that a sufficiently big iteration <math>\;\gamma^N</math> |
such that a sufficiently big iteration <math>\;\gamma^N</math> |
||
maps any given [[compact subset]] of <math>{\mathbb C}^n</math> |
maps any given [[compact subset]] of <math>{\mathbb C}^n</math> |
Revision as of 12:41, 22 June 2022
In complex geometry, a Hopf manifold (Hopf 1948) is obtained as a quotient of the complex vector space (with zero deleted) by a free action of the group of integers, with the generator of acting by holomorphic contractions. Here, a holomorphic contraction is a map such that a sufficiently big iteration maps any given compact subset of onto an arbitrarily small neighbourhood of 0.
Two-dimensional Hopf manifolds are called Hopf surfaces.
Examples
In a typical situation, is generated by a linear contraction, usually a diagonal matrix , with a complex number, . Such manifold is called a classical Hopf manifold.
Properties
A Hopf manifold is diffeomorphic to . For , 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, MR 0023054
- Ornea, Liviu (2001) [1994], "Hopf manifold", Encyclopedia of Mathematics, EMS Press