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

Not every complex analytic variety is algebraic, though.

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