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.

The Riemann sphere is one of the simplest complex algebraic varieties.

Chow's theorem

Main article: 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.

See also

• Complete variety
• Complex analytic variety

Bibliography

• 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.