Andreotti–Frankel theorem

In mathematics, the Andreotti–Frankel theorem from 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

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

and

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

This theorem applies in particular to any smooth affine variety of dimension $n$ .

References

John Willard Milnor (1963), Morse Theory, Ch. 7.