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 puts any given compact subset onto an arbitrarily small neighbourhood of 0.

Two dimensional Hopf manifolds are called Hopf surfaces.


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.


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.


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