Andreotti–Frankel theorem explained

In mathematics, the Andreotti–Frankel theorem, introduced by, states that if

V is a smooth, complex affine variety of complex dimension n or, more generally, if V is any Stein manifold of dimension n , then V admits a Morse function with critical points of index at most n, and so V is homotopy equivalent to a CW complex of real dimension at most n.

Consequently, if

V \subseteq \C ^{r} is a closed connected complex submanifold of complex dimension n , then V has the homotopy type of a CW complex of real dimension \le n .ThereforeH ^{i(V;} \Z) =0 , for i>n andH _{i(V;} \Z) =0 , for i>n. This theorem applies in particular to any smooth, complex affine variety of dimension n .

References

• • Book: Milnor , John W. . John Milnor

. John Milnor . Notes by Michael Spivak and Robert Wells . Morse theory . Annals of Mathematics Studies, No. 51 . . Princeton, NJ . 1963 . 0-691-08008-9 . Chapter 7.