Andreotti–Frankel theorem

In mathematics, the Andreotti–Frankel theorem, introduced by Aldo Andreotti and Theodore Frankel (1959), states that if ${\displaystyle V}$ is a smooth, complex affine variety of complex dimension ${\displaystyle n}$ or, more generally, if ${\displaystyle V}$ is any Stein manifold of dimension ${\displaystyle n}$, then ${\displaystyle V}$ admits a Morse function with critical points of index at most n, and so ${\displaystyle V}$ is homotopy equivalent to a CW complex of real dimension at most n.

Consequently, if ${\displaystyle V\subseteq \mathbb {C} ^{r}}$ is a closed connected complex submanifold of complex dimension ${\displaystyle n}$, then ${\displaystyle V}$ has the homotopy type of a CW complex of real dimension ${\displaystyle \leq n}$. Therefore

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

and

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

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

References

• Andreotti, Aldo; Frankel, Theodore (1959), "The Lefschetz theorem on hyperplane sections", Annals of Mathematics, Second Series, 69 (3): 713–717, doi:10.2307/1970034, ISSN 0003-486X, JSTOR 1970034, MR 0177422
• 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. {{cite book}}: ISBN / Date incompatibility (help) Chapter 7.