Complex dimension
Encyclopedia
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.
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
, complex dimension usually refers to the dimension of a complex manifold
Complex manifold
In differential geometry, a complex manifold is a manifold with an atlas of charts to the open unit disk in Cn, such that the transition maps are holomorphic....
M, or complex algebraic variety
Algebraic variety
In mathematics, an algebraic variety is the set of solutions of a system of polynomial equations. Algebraic varieties are one of the central objects of study in algebraic geometry...
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
Mathematical singularity
In mathematics, a singularity is in general a point at which a given mathematical object is not defined, or a point of an exceptional set where it fails to be well-behaved in some particular way, such as differentiability...
V will also be a smooth manifold of dimension 2d.
The same points apply to codimension
Codimension
In mathematics, codimension is a basic geometric idea that applies to subspaces in vector spaces, and also to submanifolds in manifolds, and suitable subsets of algebraic varieties.The dual concept is relative dimension.-Definition:...
. For example a smooth complex hypersurface in complex projective space
Complex projective space
In mathematics, complex projective space is the projective space with respect to the field of complex numbers. By analogy, whereas the points of a real projective space label the lines through the origin of a real Euclidean space, the points of a complex projective space label the complex lines...
of dimension n will be a manifold of dimension 2(n − 1). A complex hyperplane
Hyperplane
A hyperplane is a concept in geometry. It is a generalization of the plane into a different number of dimensions.A hyperplane of an n-dimensional space is a flat subset with dimension n − 1...
does not separate complex projective space into two components, because it has codimension 2.