algebraic manifold
{{short description|Algebraic variety}}
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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 {{nowrap|1=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.
Examples
See also
References
- {{cite journal | last=Nash | first=John Forbes | authorlink=John Forbes Nash | title = Real algebraic manifolds | journal = Annals of Mathematics | year = 1952 | pages = 405–21 | volume = 56 | issue=3 | mr=0050928 | doi=10.2307/1969649| jstor=1969649 }} (See also Proc. Internat. Congr. Math., 1950, (AMS, 1952), pp. 516–517.)
External links
- [http://planetmath.org/encyclopedia/KAlgebraicManifold.html K-Algebraic manifold at PlanetMath]
- [http://mathworld.wolfram.com/AlgebraicManifold.html Algebraic manifold at Mathworld]
- [http://www.mccme.ru/ium/postscript/s99/notes/lec-23.ps.gz Lecture notes on algebraic manifolds]
- [http://gorod.bogomolov-lab.ru/ps/stud/projgeom/1718/lec_08.pdf Lecture notes on algebraic manifolds]
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