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 ${\displaystyle \mathbb {C} \mathbf {P} ^{n}}$, is an algebraic variety. These are usually simply referred to as projective varieties.

Let X be a complex algebraic variety. Then there is a projective resolution of singularities ${\displaystyle X'\to X}$.²

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