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 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, John Forbes (1952). "Real algebraic manifolds". Annals of Mathematics. 56 (3): 405–21. doi:10.2307/1969649. JSTOR 1969649. MR 0050928. (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
  • Lecture notes on algebraic manifolds