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.^[1]

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.

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

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

^ Parshin, Alexei N., and Igor Rostislavovich Shafarevich, eds. Algebraic Geometry III: Complex Algebraic Varieties. Algebraic Curves and Their Jacobians. Vol. 3. Springer, 1998. ISBN 3-540-54681-2
^ (Abramovich 2017)

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.