Complex algebraic varietyIn algebraic geometry, a complex algebraic variety is an algebraic variety over the field of complex numbers.Chow's theoremAlgebraic geometry and [analytic geometry#Chow.27s theorem|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.Hironaka's theoremLet be a complex algebraic variety. Then there is a projective resolution of singularities.Relation with similar conceptsDespite Chow's theorem, not every complex analytic variety is a complex algebraic variety.