[1] These are spaces in which the local neighborhoods of points (or of non-singular points in the case of a variety) are modeled on a Cartesian product of the form
, and the complex dimension is the exponent
[2] That is, a smooth manifold of complex dimension
, away from any singular point, will also be a smooth manifold of real dimension
However, for a real algebraic variety (that is a variety defined by equations with real coefficients), its dimension refers commonly to its complex dimension, and its real dimension refers to the maximum of the dimensions of the manifolds contained in the set of its real points.
The real dimension is not greater than the dimension, and equals it if the variety is irreducible and has real points that are nonsingular.
defines a variety of (complex) dimension 2 (a surface), but of real dimension 0 — it has only one real point, (0, 0, 0), which is singular.
[3] The same considerations 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 a complex projective space into two components, because it has real codimension 2.
This mathematical analysis–related article is a stub.