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Andreotti–Frankel theorem

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Title: Andreotti–Frankel theorem  
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Subject: Complex manifolds, Homotopy theory, Lefschetz hyperplane theorem, List of theorems
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Andreotti–Frankel theorem

In mathematics, the Andreotti–Frankel theorem, introduced by Andreotti and Frankel (1959), states that if V is a smooth affine variety of complex dimension n or, more generally, if V is any Stein manifold of dimension n, then in fact V is homotopy equivalent to a CW complex of real dimension at most n. In other words V has only half as much topology.

Consequently, if V \subseteq \mathbb{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; \bold Z)=0,\text{ for }i>n \,

and

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

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

References

  • Andreotti, Aldo; Frankel, Theodore (1959), "The Lefschetz theorem on hyperplane sections",  
  • John Willard Milnor (1963), Morse Theory, Ch. 7.


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