Algebraic Manifold

An algebraic manifold is another name for a smooth algebraic variety. It can be covered by coordinate charts so that the transition functions are given by rational functions. Technically speaking, the coordinate charts should be to all of affine space .

For example, the sphere is an algebraic manifold, with a chart given by stereographic projection to , and another chart at , with the transition function given by . In this setting, it is called the Riemann sphere. The torus is also an algebraic manifold, in this setting called an elliptic curve, with charts given by elliptic functions such as the Weierstrass elliptic function.