Andreotti–Frankel theorem
In mathematics, the Andreotti–Frankel theorem, introduced by Aldo Andreotti and Theodore Frankel (1959), states that if [math]\displaystyle{ V }[/math] is a smooth, complex affine variety of complex dimension [math]\displaystyle{ n }[/math] or, more generally, if [math]\displaystyle{ V }[/math] is any Stein manifold of dimension [math]\displaystyle{ n }[/math], then [math]\displaystyle{ V }[/math] admits a Morse function with critical points of index at most n, and so [math]\displaystyle{ V }[/math] is homotopy equivalent to a CW complex of real dimension at most n.
Consequently, if [math]\displaystyle{ V \subseteq \C^r }[/math] is a closed connected complex submanifold of complex dimension [math]\displaystyle{ n }[/math], then [math]\displaystyle{ V }[/math] has the homotopy type of a CW complex of real dimension [math]\displaystyle{ \le n }[/math]. Therefore
- [math]\displaystyle{ H^i(V; \Z)=0,\text{ for }i\gt n }[/math]
and
- [math]\displaystyle{ H_i(V; \Z)=0,\text{ for }i\gt n. }[/math]
This theorem applies in particular to any smooth, complex affine variety of dimension [math]\displaystyle{ n }[/math].
References
- Andreotti, Aldo; Frankel, Theodore (1959), "The Lefschetz theorem on hyperplane sections", Annals of Mathematics, Second Series 69: 713–717, doi:10.2307/1970034, ISSN 0003-486X
- Milnor, John W. (1963). Morse theory. Annals of Mathematics Studies, No. 51. Notes by Michael Spivak and Robert Wells. Princeton, NJ: Princeton University Press. ISBN 0-691-08008-9. Chapter 7.
 |  |
