In algebraic geometry, a complex algebraic variety is an

algebraic variety Algebraic varieties are the central objects of study in algebraic geometry, a sub-field of mathematics. Classically, an algebraic variety is defined as the set of solutions of a system of polynomial equations over the real or complex numbers ...

(in the scheme sense or otherwise) over the field of

complex number In mathematics, a complex number is an element of a number system that extends the real numbers with a specific element denoted , called the imaginary unit and satisfying the equation i^= -1; every complex number can be expressed in the for ...

s. Parshin, Alexei N., and

Igor Rostislavovich Shafarevich Igor Rostislavovich Shafarevich (russian: И́горь Ростисла́вович Шафаре́вич; 3 June 1923 – 19 February 2017) was a Soviet and Russian mathematician who contributed to algebraic number theory and algebraic geometry ...

, eds. ''Algebraic Geometry III: Complex Algebraic Varieties. Algebraic Curves and Their Jacobians.'' Vol. 3. Springer, 1998.

Chow's theorem

Chow's theorem states that a projective analytic variety; i.e., a closed analytic subvariety of the

complex projective space In mathematics, complex projective space is the projective space with respect to the field of complex numbers. By analogy, whereas the points of a real projective space label the lines through the origin of a real Euclidean space, the points of ...

\mathbb\mathbf^n

is an algebraic variety; it is usually simply referred to as a

projective variety In algebraic geometry, a projective variety over an algebraically closed field ''k'' is a subset of some projective ''n''-space \mathbb^n over ''k'' that is the zero-locus of some finite family of homogeneous polynomials of ''n'' + 1 variables ...

Relation with similar concepts

Not every complex analytic variety is algebraic, though.

References

{{reflist Algebraic varieties