Algebraic manifold

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. An example is the sphere, which can be defined as the zero set of the polynomial x² + y² + z² – 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

See also

• Algebraic geometry and analytic geometry

References

• Nash, J. Real algebraic manifolds. (1952) Ann. Math. 56 (1952), 405–421. (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