called algebraic spaces and stacks. Algebraic manifolds An algebraic manifold is an algebraic variety that is also an m-dimensional manifold, and hence every sufficiently small local patch is isomorphic to km. Equivalently, the variety is smooth (free from singular points). When is the real numbers, R, algebraic manifolds are called Nash manifolds. Algebraic manifolds can be defined as the zero set of a finite collection of analytic algebraic functions. Projective algebraic manifolds are an equivalent definition for projective varieties. The Riemann sphere is one example. See also Variety (disambiguation) — listing also several mathematical meanings Function field of an algebraic variety Birational geometry Motive (algebraic geometry) Analytic variety Zariski–Riemann space Semi-algebraic set Fano variety Notes References Sources Milne J., Jacobian Varieties, published as Chapter VII of Arithmetic geometry (Storrs, Conn., 1984), 167–212, Springer, New York, 1986. Algebraic geometry