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 $$x^2 + y^2 + z^2 – 1$$, 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.
Examples

Elliptic curves
Grassmannian

Algebraic geometry and analytic geometry

References

Nash, John Forbes (1952). "Real algebraic manifolds". Annals of Mathematics. 56 (3): 405–21. doi:10.2307/1969649. MR 0050928. (See also Proc. Internat. Congr. Math., 1950, (AMS, 1952), pp. 516–517.)



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