In complex geometry, a Hopf surface is a compact complex surface obtained as a quotient of the complex vector space by a free action of a discrete group. If this group is the integers the Hopf surface is called primary, otherwise it is called secondary. (Some authors use the term "Hopf surface" to mean "primary Hopf surface".) The first example was found by by multiplication by 2; this was the first example of a compact complex surface with no Kähler metric.

Higher-dimensional analogues of Hopf surfaces are called Hopf manifolds.

Invariants

Hopf surfaces are surfaces of class VII and in particular all have Kodaira dimension, and all their plurigenera vanish. The geometric genus is 0. The fundamental group has a normal central infinite cyclic subgroup of finite index. The Hodge diamond is In particular the first Betti number is 1 and the second Betti number is 0. Conversely showed that a compact complex surface with vanishing the second Betti number and whose fundamental group contains an infinite cyclic subgroup of finite index is a Hopf surface.

Primary Hopf surfaces

In the course of classification of compact complex surfaces, Kodaira classified the primary Hopf surfaces.

A primary Hopf surface is obtained as

can be written as

are complex numbers.

These surfaces contain an elliptic curve (the image of the x the image of the y for some positive integers m and n, and otherwise the only curves are the two images of the axes.

The Picard group.

if and only if it is a primary Hopf surface.

Secondary Hopf surfaces

Any secondary Hopf surface has a finite unramified cover that is a primary Hopf surface. Equivalently, its fundamental group has a subgroup of finite index in its center that is isomorphic to the integers. classified them by finding the finite groups acting without fixed points on primary Hopf surfaces.

Many examples of secondary Hopf surfaces can be constructed with underlying space a product of a spherical space forms and a circle.

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