Algebraic manifold
# Algebraic manifold

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

* Elliptic curves
* Grassmannian
