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]

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

Chow's theorem

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

Hironaka's theorem

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

Relation with similar concepts

Not every complex analytic variety is algebraic, though.

References

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)

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.

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[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)