Hopf manifold

In complex geometry, a Hopf manifold (Hopf 1948) is obtained as a quotient of the complex vector space (with zero deleted) ${\displaystyle ({\mathbb {C} }^{n}\backslash 0)}$ by a free action of the group ${\displaystyle \Gamma \cong {\mathbb {Z} }}$ of integers, with the generator ${\displaystyle \gamma }$ of ${\displaystyle \Gamma }$ acting by holomorphic contractions. Here, a holomorphic contraction is a map ${\displaystyle \gamma$ :\;{\mathbb {C} }^{n}\mapsto {\mathbb {C} }^{n}} such that a sufficiently big iteration ${\displaystyle \;\gamma ^{N}}$ maps any given compact subset of ${\displaystyle {\mathbb {C} }^{n}}$ onto an arbitrarily small neighbourhood of 0.

Two-dimensional Hopf manifolds are called Hopf surfaces.

Examples

In a typical situation, ${\displaystyle \Gamma }$ is generated by a linear contraction, usually a diagonal matrix ${\displaystyle q\cdot Id}$, with ${\displaystyle q\in {\mathbb {C} }}$ a complex number, ${\displaystyle 0<|q|<1}$. Such manifold is called a classical Hopf manifold.

Properties

A Hopf manifold ${\displaystyle H:=({\mathbb {C} }^{n}\backslash 0)/{\mathbb {Z} }}$ is diffeomorphic to ${\displaystyle S^{2n-1}\times S^{1}}$. For ${\displaystyle n\geq 2}$, 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