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 2d. That is, the smooth manifold M has dimension 2d; and away from any singular points V will also be a smooth manifold of dimension 2d.
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.
Famous quotes containing the words complex and/or dimension:
“All of life and human relations have become so incomprehensibly complex that, when you think about it, it becomes terrifying and your heart stands still.”
—Anton Pavlovich Chekhov (1860–1904)
“Le Corbusier was the sort of relentlessly rational intellectual that only France loves wholeheartedly, the logician who flies higher and higher in ever-decreasing circles until, with one last, utterly inevitable induction, he disappears up his own fundamental aperture and emerges in the fourth dimension as a needle-thin umber bird.”
—Tom Wolfe (b. 1931)