PlanetMath: complex line

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complex line

(Definition)

Definition 1 Let $a, b \in {\mathbb{C}}^n$ . The set $\ell := \{a + b z \mid z \in {\mathbb{C}} \}$ is called the complex line.

A complex line is a holomorphic complex affine imbedding of ${\mathbb{C}}$ into ${\mathbb{C}}^n$ so that if is holomorphic, then $z \mapsto f(a + b z)$ 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 2 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

$\displaystyle \{a + \sum_{j=1}^k b_k z_k \mid z_1,\ldots,z_k \in {\mathbb{C}} \}$

is called the complex affine space.

Bibliography

1: Steven G. Krantz. Function Theory of Several Complex Variables, AMS Chelsea Publishing, Providence, Rhode Island, 1992.

"complex line" is owned by jirka.

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