Andreotti–Frankel theorem
Mathematical theorem of complex manifolds / From Wikipedia, the free encyclopedia
In mathematics, the Andreotti–Frankel theorem, introduced by Aldo Andreotti and Theodore Frankel (1959), 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 .