Complex dimension

Article summary

In mathematics, complex dimension usually refers to the dimension of a complex manifold or a complex algebraic variety. 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 \mathbb{C}^d for some d , and the complex dimension is the exponent d in this product. Because \mathbb{C} can in turn be modeled by \mathbb{R}^2 , a space with complex dimension d will have real dimension 2d . That is, a smooth manifold of complex dimension d has real dimension 2d ; and a complex algebraic variety of complex dimension d , away from any singular point, will also be a smooth manifold of real dimension 2d . 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. For example, the equation x^2+y^2+z^2=0 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. The same considerations apply to codimension.…

Key claims

No claims recorded.

Discussion questions

1. Summarise the article in your own words. Which single claim would you verify first, and why?
2. Which claim in this article is most likely to be revised as the sources update? How would you check its current status?
3. If two sources disagreed about one of these facts, what would you need in order to trust one over the other?

Bibliography

No sources recorded.