Complex algebraic variety

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 states that a projective complex analytic variety, i.e., a closed analytic subvariety of the complex projective space

These are usually simply referred to as projective varieties.

Let X be a complex algebraic variety.

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