Andreotti–Frankel theorem

In mathematics, the Andreotti–Frankel theorem, introduced by, 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 $$V \subseteq \C^r$$ 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
 * $$H^i(V; \Z)=0,\text{ for }i>n $$

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

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