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 $\gamma :\;{\mathbb {C} }^{n}\to {\mathbb {C} }^{n}$ 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