Andreotti–Frankel theorem
# Andreotti–Frankel 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 $n$.

