In mathematics, the Andreotti–Frankel theorem, introduced by Andreotti and Frankel (1959), states that if is a smooth affine variety of complex dimension or, more generally, if is any Stein manifold of dimension , then in fact is homotopy equivalent to a CW complex of real dimension at most n. In other words has only half as much topology.
Consequently, if is a closed connected complex submanifold of complex dimension , then has the homotopy type of a complex of real dimension . Therefore
This theorem applies in particular to any smooth affine variety of dimension .
- Andreotti, Aldo; Frankel, Theodore (1959), "The Lefschetz theorem on hyperplane sections", Annals of Mathematics. Second Series, 69: 713–717, ISSN 0003-486X, JSTOR 1970034, MR 0177422
- John Willard Milnor (1963), Morse Theory, Ch. 7.
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