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 $n$.

References

• Andreotti, Aldo; Frankel, Theodore (1959), "The Lefschetz theorem on hyperplane sections", Annals of Mathematics, Second Series, 69: 713–717, doi:10.2307/1970034, ISSN 0003-486X, JSTOR 1970034, MR 0177422
• Milnor, John W. (1963). Morse theory. Annals of Mathematics Studies, No. 51. Notes by Michael Spivak and Robert Wells. Princeton, NJ: Princeton University Press. ISBN 0-691-08008-9. Chapter 7.