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In
complex geometry In mathematics, complex geometry is the study of geometric structures and constructions arising out of, or described by, the complex numbers. In particular, complex geometry is concerned with the study of spaces such as complex manifolds and co ...
, a Hopf manifold is obtained as a quotient of the complex
vector space In mathematics and physics, a vector space (also called a linear space) is a set whose elements, often called '' vectors'', may be added together and multiplied ("scaled") by numbers called '' scalars''. Scalars are often real numbers, but ...
(with zero deleted) (^n\backslash 0) by a
free action In mathematics, a group action on a space is a group homomorphism of a given group into the group of transformations of the space. Similarly, a group action on a mathematical structure is a group homomorphism of a group into the automorphism ...
of the group \Gamma \cong of
integer An integer is the number zero (), a positive natural number (, , , etc.) or a negative integer with a minus sign ( −1, −2, −3, etc.). The negative numbers are the additive inverses of the corresponding positive numbers. In the language ...
s, with the generator \gamma of \Gamma acting by holomorphic contractions. Here, a ''holomorphic contraction'' is a map \gamma:\; ^n \to ^n such that a sufficiently big iteration \;\gamma^N maps any given
compact subset In mathematics, specifically general topology, compactness is a property that seeks to generalize the notion of a closed and bounded subset of Euclidean space by making precise the idea of a space having no "punctures" or "missing endpoints", i ...
of ^n onto an arbitrarily small
neighbourhood A neighbourhood (British English, Irish English, Australian English and Canadian English) or neighborhood (American English; American and British English spelling differences, see spelling differences) is a geographically localised community ...
of 0. Two-dimensional Hopf manifolds are called
Hopf surface In complex geometry, a Hopf surface is a compact complex surface obtained as a quotient of the complex vector space (with zero deleted) \Complex^2\setminus \ by a free action of a discrete group. If this group is the integers the Hopf surface is cal ...
s.


Examples

In a typical situation, \Gamma is generated by a linear contraction, usually a
diagonal matrix In linear algebra, a diagonal matrix is a matrix in which the entries outside the main diagonal are all zero; the term usually refers to square matrices. Elements of the main diagonal can either be zero or nonzero. An example of a 2×2 diagonal ...
q\cdot Id, with q\in a complex number, 0<, q, <1. Such manifold is called ''a classical Hopf manifold''.


Properties

A Hopf manifold H:=(^n\backslash 0)/ is
diffeomorphic In mathematics, a diffeomorphism is an isomorphism of smooth manifolds. It is an invertible function that maps one differentiable manifold to another such that both the function and its inverse are differentiable. Definition Given two ...
to S^\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

* *{{SpringerEOM, title=Hopf manifold, first=Liviu , last=Ornea Complex manifolds