Hopf manifold

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 $\;\gamma ^{N}$ maps any given compact subset of ${\mathbb {C} }^{n}$ 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. 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