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''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|Analytic surface]].