Hopf manifold

 ( Complex Manifolds ) 

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

Two-dimensional Hopf manifolds are called .

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

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Properties A Hopf manifold $H:=(\{\backslash mathbb\; C\}^n\backslash backslash\; 0)/\{\backslash mathbb\; Z\}$ is diffeomorphic to $S^\{2n-1\}\backslash times\; S^1$. For $n\backslash geq\; 2$, it is non-Kähler. In fact, it is not even symplectic because the second cohomology group is zero.

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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.

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