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) C² \ 0 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 Hopf (1948), with the discrete group isomorphic to the integers, with a generator acting on C² 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.

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 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.

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

A primary Hopf surface is obtained as

where ${\displaystyle \Gamma }$ is a group generated by a polynomial contraction ${\displaystyle \gamma }$. Kodaira has found a normal form for ${\displaystyle \gamma }$. In appropriate coordinates, ${\displaystyle \gamma }$ can be written as

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