In mathematics, complex dimension usually refers to the dimension of a complex manifold M , or complex algebraic variety V . If the complex dimension is d , the real dimension will be 2 d . That is, the smooth manifold M has dimension 2 d ; and away from any singular points V will also be a smooth manifold of dimension 2 d .

The same points apply to Codimension. For example a smooth complex hypersurface in complex projective space of dimension n will be a manifold of dimension 2( n − 1). A complex hyperplane does not separate complex projective space into two components, because it has codimension 2.