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 states that a projective complex analytic variety, i.e., a closed analytic subvariety of the complex projective space , is an algebraic variety. These are usually simply referred to as projective varieties.
Let be a complex algebraic variety. Then there is a projective resolution of singularities .
Despite Chow's theorem, not every complex analytic variety is a complex algebraic variety.