 Andreotti-Frankel Theorem
Get Andreotti%E2%80%93Frankel Theorem essential facts below. View Videos or join the Andreotti%E2%80%93Frankel Theorem discussion. Add Andreotti%E2%80%93Frankel Theorem to your PopFlock.com topic list for future reference or share this resource on social media.
Andreotti%E2%80%93Frankel Theorem

In mathematics, the Andreotti-Frankel theorem, introduced by Aldo Andreotti and Theodore Frankel (1959), 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 $V\subseteq \mathbb {C} ^{r}$ is a closed connected complex submanifold of complex dimension $n$ , then $V$ has the homotopy type of a CW complex of real dimension $\leq n$ . Therefore

$H^{i}(V;\mathbb {Z} )=0,{\text{ for }}i>n$ and

$H_{i}(V;\mathbb {Z} )=0,{\text{ for }}i>n.$ This theorem applies in particular to any smooth, complex affine variety of dimension $n$ .