In algebraic geometry, a complex algebraic variety is an algebraic variety (in the scheme sense or otherwise) over the field of complex numbers.^[1]

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

Let X be a complex algebraic variety. Then there is a projective resolution of singularities ${\displaystyle X'\to X}$.^[2]

Despite Chow's theorem, not every complex analytic variety is a complex algebraic variety.

• Complete variety
• Complex analytic variety

• Abramovich, Dan (2017). "Resolution of singularities of complex algebraic varieties and their families". Proceedings of the International Congress of Mathematicians (ICM 2018). pp. 523–546. arXiv:1711.09976. doi:10.1142/9789813272880_0066. ISBN 978-981-327-287-3. S2CID 119708681.
• Hironaka, Heisuke (1964). "Resolution of Singularities of an Algebraic Variety over a Field of Characteristic Zero: I". Annals of Mathematics. 79 (1): 109–203. doi:10.2307/1970486. JSTOR 1970486.