Analytic plane

complex-analytic plane

A non-empty set of points $ z = ( z _ {1} \dots z _ {n} ) $ in the complex vector space $ \mathbf C ^ {n} $ that satisfies a system of equations

$$ \sum _ {i = 1 } ^ { n } a _ {i j } z _ {i} = b _ {j} , \ j = 1 \dots k ; \ a _ {i j } , b _ {j} \in \mathbf C ; $$

$$ \mathop{\rm rank} \| a _ {i j } \| = k < n . $$

The number $ k $ is called the complex codimension, while $ n - k $ is called the complex dimension of the analytic plane. The real dimension of the analytic plane equals $ 2 (n - k ) $ and is even, but not all even-dimensional real planes in $ \mathbf R ^ {2n} = \mathbf C ^ {n} $ are analytic planes. Complex one-dimensional analytic planes are sometimes called complex, or analytic, straight lines. See also Analytic surface.