Complex algebraic variety

The Riemann sphere is one of the simplest complex algebraic varieties.

In algebraic geometry, a complex algebraic variety is an algebraic variety (in the scheme sense or otherwise) over the field of complex numbers.

Chow's theorem

Main article: Chow's theorem

Chow's theorem states that a projective complex analytic variety, i.e., a closed analytic subvariety of the complex projective space $ℂ{𝐏}^n$, is an algebraic variety. These are usually simply referred to as projective varieties.

Hironaka's theorem

Let X be a complex algebraic variety. Then there is a projective resolution of singularities $X^{\prime }\to X$.

Relation with similar concepts

Despite Chow's theorem, not every complex analytic variety is a complex algebraic variety.

See also

• Complete variety
• Complex analytic variety