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| In [[complex geometry]], a '''Hopf manifold''' {{harv|Hopf|1948}} is obtained
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| as a quotient of the complex [[vector space]]
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| (with zero deleted) <math>({\Bbb C}^n\backslash 0)</math>
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| by a [[Group action|free action]] of the [[Group (mathematics)|group]] <math>\Gamma \cong {\Bbb Z}</math> of
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| [[integer]]s, with the generator <math>\gamma</math>
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| of <math>\Gamma</math> acting by holomorphic [[Contraction mapping|contractions]]. Here, a ''holomorphic contraction''
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| is a map <math>\gamma:\; {\Bbb C}^n \mapsto {\Bbb C}^n</math> | |
| such that a sufficiently big iteration <math>\;\gamma^N</math>
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| puts any given [[compact subset]] <math>{\Bbb C}^n</math>
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| onto an arbitrarily small [[Neighbourhood (mathematics)|neighbourhood]] of 0.
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| Two dimensional Hopf manifolds are called [[Hopf surface]]s.
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| == Examples ==
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| In a typical situation, <math>\Gamma</math> is generated
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| by a linear contraction, usually a [[diagonal matrix]]
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| <math>q\cdot Id</math>, with <math>q\in {\Bbb C}</math>
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| a complex number, <math>0<|q|<1</math>. Such manifold
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| is called ''a classical Hopf manifold''. | |
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| == Properties ==
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| A Hopf manifold <math>H:=({\Bbb C}^n\backslash 0)/{\Bbb Z}</math>
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| is [[diffeomorphic]] to <math>S^{2n-1}\times S^1</math>.
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| For <math>n\geq 2</math>, it is non-[[Kähler manifold|Kähler]]. Indeed,
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| the first [[cohomology group]] of ''H''
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| is odd-dimensional. By [[Hodge decomposition]],
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| odd cohomology of a compact [[Kähler manifold]]
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| are always even-dimensional.
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| == Hypercomplex structure ==
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| Even-dimensional Hopf manifolds admit
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| [[Hypercomplex manifold|hypercomplex structure]].
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| The Hopf surface is the only compact [[hypercomplex manifold]] of quaternionic dimension 1 which is not [[hyperkähler manifold|hyperkähler]].
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| == References ==
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| *{{Citation | last1=Hopf | first1=Heinz | author1-link=Heinz Hopf | title=Studies and Essays Presented to R. Courant on his 60th Birthday, January 8, 1948 | publisher=Interscience Publishers, Inc., New York | id={{MathSciNet | id = 0023054}} | year=1948 | chapter=Zur Topologie der komplexen Mannigfaltigkeiten | pages=167–185}}
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| *{{eom|id=H/h110270|first=L. |last=Ornea}}
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| [[Category:Complex manifolds]]
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They call me Emilia. Puerto Rico is where he and his wife live. Hiring is my occupation. To gather cash is what his family and him enjoy.
my website; at home std test (visit web site)