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 $V \subseteq C^r$ is a closed connected complex submanifold of complex dimension $n$. Then $V$ has the homotopy type of a $CW$ complex of real dimension $\le n$. Therefore

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

and

$H_i(V; \bold Z)=0,\text{ for }i>n. \,$

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, MR0177422, http://www.jstor.org/stable/1970034 
• John Willard Milnor (1963), Morse Theory, Ch. 7.