# complex line

###### Definition.

Let $a,b\in {\u2102}^{n}$. The set $\mathrm{\ell}:=\{a+bz\mid z\in \u2102\}$ is called the complex line.

A complex line is a holomorphic complex affine imbedding of $\u2102$ into ${\u2102}^{n}$ so that if $f$ is holomorphic, then $z\mapsto f(a+bz)$ is also holomorphic. That is the complex structures of $\mathrm{\ell}$ and ${\u2102}^{n}$ are compatible. Hence not every two dimensional real affine space is a complex line.

###### Definition.

Let $a,{b}_{1},\mathrm{\dots},{b}_{k}\in {\u2102}^{n}$ such that
${b}_{1},\mathrm{\dots},{b}_{k}$ are linearly independent^{}
over $\u2102$,
then. The set

$$\{a+\sum _{j=1}^{k}{b}_{k}{z}_{k}\mid {z}_{1},\mathrm{\dots},{z}_{k}\in \u2102\}$$ |

is called the complex affine space.

## References

- 1 Steven G. Krantz. , AMS Chelsea Publishing, Providence, Rhode Island, 1992.

Title | complex line |
---|---|

Canonical name | ComplexLine |

Date of creation | 2013-03-22 14:29:05 |

Last modified on | 2013-03-22 14:29:05 |

Owner | jirka (4157) |

Last modified by | jirka (4157) |

Numerical id | 5 |

Author | jirka (4157) |

Entry type | Definition |

Classification | msc 32-00 |

Related topic | AffineTransformation |

Defines | complex affine space |