Algebraic manifold

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Contents 1 Algebraic manifolds 2 Examples 3 See also 4 References 5 External links

[edit] Algebraic manifolds

Algebraic manifolds are an algebraic variety which are also m-dimensional manifolds. They 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.

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). Algebraic manifolds over the field k = R of real numbers are sometimes called Nash manifolds. Algebraic manifolds can be defined as the zero set of a finite collection of analytic algebraic functions. The notion of projective variety can be equivalently defined as projective algebraic manifold. The Riemann sphere is one example.

[edit] Examples

• Elliptic curves
• Grassmannian

[edit] See also

• Projective algebraic manifold
• Elliptic curve
• Algebraic geometry and analytic geometry

[edit] 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.)

[edit] External links

• K-Algebraic manifold at PlanetMath
• Algebraic manifold at Mathworld
• Lecture notes on Algebraic mainfolds