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.^[1]

Chow's theorem

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

Relation with similar concepts

Not every complex analytic variety is algebraic, though.

References

1. Parshin, Alexei N., and Igor Rostislavovich Shafarevich, eds. Algebraic Geometry III: Complex Algebraic Varieties. Algebraic Curves and Their Jacobians. Vol. 3. Springer, 1998.