Algebraic manifold

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 km 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.

Contents

Examples

See also

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.)

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