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) ${\displaystyle \mathbb {C} ^{2}\setminus \{0\}}$ 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 Heinz Hopf (1948), with the discrete group isomorphic to the integers, with a generator acting on ${\displaystyle \mathbb {C} ^{2}}$ 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.