# complex line

###### Definition.

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

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

###### Definition.

Let $a,b_{1},\ldots,b_{k}\in{\mathbb{C}}^{n}$ such that $b_{1},\ldots,b_{k}$ are linearly independent  over ${\mathbb{C}}$, then. The set

 $\{a+\sum_{j=1}^{k}b_{k}z_{k}\mid z_{1},\ldots,z_{k}\in{\mathbb{C}}\}$

is called the complex affine space.

## References

• 1 Steven G. Krantz. , AMS Chelsea Publishing, Providence, Rhode Island, 1992.
Title complex line ComplexLine 2013-03-22 14:29:05 2013-03-22 14:29:05 jirka (4157) jirka (4157) 5 jirka (4157) Definition msc 32-00 AffineTransformation complex affine space