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.
Chow's theorem.
Chow's theorem states that a projective analytic variety; i.e., a closed analytic subvariety of the complex projective space formula_1 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.