Andreotti–Frankel theorem

Template:Short description In mathematics, the Andreotti–Frankel theorem, introduced by Template:Harvs, 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

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

and

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

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

References

• Template:Citation
• Template:Cite book Chapter 7.