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complex line
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(Definition)
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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 $f$ 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 \begin{equation*} \{a + \sum_{j=1}^k b_k z_k \mid z_1,\ldots,z_k \in {\mathbb{C}} \} \end{equation*}is called the complex affine space.
- 1
- Steven G. Krantz. Function Theory of Several Complex Variables, AMS Chelsea Publishing, Providence, Rhode Island, 1992.
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"complex line" is owned by jirka.
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Cross-references: linearly independent, affine space, real, compatible, complex structures, imbedding, complex, holomorphic
There are 4 references to this entry.
This is version 2 of complex line, born on 2004-07-23, modified 2005-03-05.
Object id is 6018, canonical name is ComplexLine.
Accessed 2922 times total.
Classification:
| AMS MSC: | 32-00 (Several complex variables and analytic spaces :: General reference works ) |
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Pending Errata and Addenda
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