Andreotti–Frankel theorem

In mathematics, the Andreotti–Frankel theorem 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

:, for $i>n$

and

:, for $i>n$.

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