Complex algebraic variety

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

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

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 ${\displaystyle \mathbb {C} \mathbf {P} ^{n}}$, 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 ${\displaystyle X'\to X}$. ^[2]

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

References

1. 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
2. ( Abramovich 2017 )

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.