by a free action of a discrete group.
The first example was found by Heinz Hopf (1948), with the discrete group isomorphic to the integers, with a generator acting on
by multiplication by 2; this was the first example of a compact complex surface with no Kähler metric.
The fundamental group has a normal central infinite cyclic subgroup of finite index.
Conversely Kunihiko Kodaira (1968) 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.
A primary Hopf surface is obtained as where
is a group generated by a polynomial contraction
Kodaira has found a normal form for
These surfaces contain an elliptic curve (the image of the x-axis) and if
the image of the y-axis is a second elliptic curve.
, the Hopf surface is an elliptic fiber space over the projective line if
for some positive integers m and n, with the map to the projective line given by
The Picard group of any primary Hopf surface is isomorphic to the non-zero complex numbers
Kodaira (1966b) has proven that a complex surface is diffeomorphic to
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.
Masahido Kato (1975) 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.