In complex geometry, a Hopf manifold (Hopf 1948) is obtained as a quotient of the complex vector space (with zero deleted) ( C n ∖ 0 ) by a free action of the group Γ ≅ Z of integers, with the generator γ of Γ acting by holomorphic contractions. Here, a holomorphic contraction is a map γ : C n ↦ C n such that a sufficiently big iteration γ N puts any given compact subset C n onto an arbitrarily small neighbourhood of 0.
Two dimensional Hopf manifolds are called Hopf surfaces.
In a typical situation, Γ is generated by a linear contraction, usually a diagonal matrix q ⋅ I d , with q ∈ C a complex number, 0 < | q | < 1 . Such manifold is called a classical Hopf manifold.
A Hopf manifold H := ( C n ∖ 0 ) / Z is diffeomorphic to S 2 n − 1 × S 1 . For n ≥ 2 , it is non-Kähler. In fact, it is not even symplectic because the second cohomology group is zero.
Even-dimensional Hopf manifolds admit hypercomplex structure. The Hopf surface is the only compact hypercomplex manifold of quaternionic dimension 1 which is not hyperkähler.
