Andreotti–Frankel theorem

In mathematics, the Andreotti–Frankel theorem, introduced by Andreotti and Frankel (1959), states that if V is a smooth affine variety of complex dimension n or, more generally, if V is any Stein manifold of dimension n , then in fact V is homotopy equivalent to a CW complex of real dimension at most n. In other words V has only half as much topology.

Consequently, if V ⊆ C r is a closed connected complex submanifold of complex dimension n , then V has the homotopy type of a C W complex of real dimension ≤ n . Therefore

and

This theorem applies in particular to any smooth affine variety of dimension n .