nLab complex line

Idea

A complex line is a complex vector space of dimension 1 (of complex dimension that is, as a $\mathbb{C}$-vector space, meaning that as a vector space over the real numbers it is a plane).

In particular the complex plane $\mathbb{C}$ itself is a complex line; conversely, any two complex lines are isomorphic (to each other and to the complex plane). However, there are many such isomorphisms; the automorphism group is $\mathbb{C} \setminus \{0\}$.

Related concepts

• real line, continuum

• complex vector space

  • complex line

• complex vector bundle

  • complex line bundle