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 complex analytic variety, i.e., a closed analytic subvariety of the complex projective space $$\mathbb{C}\mathbf{P}^n$$, is an algebraic variety. These are usually simply referred to as projective varieties.

Hironaka's theorem
Let $X$ be a complex algebraic variety. Then there is a projective resolution of singularities $$X' \to X$$.

Relation with similar concepts
Despite Chow's theorem, not every complex analytic variety is a complex algebraic variety.