Andreotti–Frankel theorem
In mathematics, the Andreotti–Frankel theorem, introduced by, states that if is a smooth, complex affine variety of complex dimension or, more generally, if is any Stein manifold of dimension, then
admits a Morse function with critical points of index at most n, and so is homotopy equivalent to a CW complex of real dimension at most n.
Consequently, if is a closed connected complex submanifold of complex dimension, then has the homotopy type of a CW complex of real dimension.
Therefore
and
This theorem applies in particular to any smooth, complex affine variety of dimension.
