Andreotti–Frankel theorem

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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 $C W$ complex of real dimension $\le n$ . Therefore