In mathematics, the Andreotti–Frankel theorem, introduced by Aldo Andreotti and Theodore Frankel (1959), states that if V {\displaystyle V} is a smooth, complex affine variety of complex dimension n {\displaystyle n} or, more generally, if V {\displaystyle V} is any Stein manifold of dimension n {\displaystyle n} , then
V {\displaystyle V} admits a Morse function with critical points of index at most n, and so V {\displaystyle V} is homotopy equivalent to a CW complex of real dimension at most n. Consequently, if V ⊆ C r {\displaystyle V\subseteq \mathbb {C} ^{r}} is a closed connected complex submanifold of complex dimension n {\displaystyle n} , then V {\displaystyle V} has the homotopy type of a CW complex of real dimension ≤ n {\displaystyle \leq n} . Therefore
H i ( V ; Z ) = 0 , for i > n {\displaystyle H^{i}(V;\mathbb {Z} )=0,{\text{ for }}i>n}
and
H i ( V ; Z ) = 0 , for i > n . {\displaystyle H_{i}(V;\mathbb {Z} )=0,{\text{ for }}i>n.}
This theorem applies in particular to any smooth, complex affine variety of dimension n {\displaystyle n} .
References Andreotti, Aldo; Frankel, Theodore (1959), "The Lefschetz theorem on hyperplane sections", Annals of Mathematics, Second Series, 69 (3): 713–717, doi:10.2307/1970034, ISSN 0003-486X, JSTOR 1970034, MR 0177422 Milnor, John W. (1963). Morse theory. Annals of Mathematics Studies, No. 51. Notes by Michael Spivak and Robert Wells. Princeton, NJ: Princeton University Press. Reprinted with corrections, 1969, ISBN 0-691-08008-9. Chapter 7.