# Andreotti–Frankel theorem

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

Consequently, if ${\displaystyle V\subseteq \mathbb {C} ^{r}}$ is a closed connected complex submanifold of complex dimension ${\displaystyle n}$, then ${\displaystyle V}$ has the homotopy type of a ${\displaystyle CW}$ complex of real dimension ${\displaystyle \leq n}$. Therefore

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

and

${\displaystyle H_{i}(V;{\mathbf {Z}})=0,{\text{ for }}i>n.\,}$

This theorem applies in particular to any smooth affine variety of dimension ${\displaystyle n}$.