# Algebraic manifold

__: Algebraic variety__

**Short description**

In mathematics, an **algebraic manifold** is an algebraic variety which is also a manifold. As such, algebraic manifolds are a generalisation of the concept of smooth curves and surfaces defined by polynomials. An example is the sphere, which can be defined as the zero set of the polynomial *x*^{2} + *y*^{2} + *z*^{2} – 1, and hence is an algebraic variety.

For an algebraic manifold, the ground field will be the real numbers or complex numbers; in the case of the real numbers, the manifold of real points is sometimes called a Nash manifold.

Every sufficiently small local patch of an algebraic manifold is isomorphic to *k*^{m} where *k* is the ground field. Equivalently the variety is smooth (free from singular points). The Riemann sphere is one example of a complex algebraic manifold, since it is the complex projective line.

## Examples

## See also

## References

- Nash, John Forbes (1952). "Real algebraic manifolds".
*Annals of Mathematics***56**(3): 405–21. doi:10.2307/1969649. (See also Proc. Internat. Congr. Math., 1950, (AMS, 1952), pp. 516–517.)

## External links

- K-Algebraic manifold at PlanetMath
- Algebraic manifold at Mathworld
- Lecture notes on algebraic manifolds

Original source: https://en.wikipedia.org/wiki/Algebraic manifold.
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