complex line

Definition.

Let $a,b\in {ℂ}^n$. The set$\mathrm{\ell }:=\{a+bz\mid z\in ℂ\}$ is called the complex line.

A complex line is a holomorphic complex affine imbedding of $ℂ$into ${ℂ}^n$ so that if $f$ is holomorphic, then$z\mapsto f(a+bz)$ is also holomorphic. That is the complex structures of $\mathrm{\ell }$ and ${ℂ}^n$ are compatible. Hence not every two dimensional real affine space is a complex line.

Definition.

Let $a,b_1,\mathrm{\dots },b_k\in {ℂ}^n$ such that$b_1,\mathrm{\dots },b_k$ are linearly independentover $ℂ$,then. The set

$$\{a+\sum _{j=1}^k{b}_k{z}_k\mid z_1,\mathrm{\dots },z_k\in ℂ\}$$

is called the complex affine space.