Hopf manifold
In complex geometry, a Hopf manifold (Hopf 1948) is obtained as a quotient of the complex vector space (with zero deleted) by a
free action of the group
of
integers, with the generator
of acting by holomorphic contractions. Here, a holomorphic contraction
is a map
such that a sufficiently big iteration
maps any given compact subset
of
onto an arbitrarily small neighbourhood of 0.
Two-dimensional Hopf manifolds are called Hopf surfaces.
Examples
In a typical situation, is generated by a linear contraction, usually a diagonal matrix , with a complex number, . Such manifold is called a classical Hopf manifold.
Properties
A Hopf manifold is
diffeomorphic
to .
For , 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
- MR 0023054
- Ornea, Liviu (2001) [1994], "Hopf manifold", Encyclopedia of Mathematics, EMS Press