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

References

• Andreotti, Aldo; Frankel, Theodore (1959), "The Lefschetz theorem on hyperplane sections", Annals of Mathematics. Second Series 69: 713–717, ISSN 0003-486X, JSTOR 1970034, MR 0177422
• John Willard Milnor (1963), Morse Theory, Ch. 7.