In mathematics

Mathematics

Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...

, the Andreotti–Frankel theorem from 1959 states that if is a smooth affine variety of complex dimension

Complex dimension

In mathematics, complex dimension usually refers to the dimension of a complex manifold M, or complex algebraic variety V. If the complex dimension is d, the real dimension will be 2d...

  or, more generally, if is any Stein manifold

Stein manifold

In mathematics, a Stein manifold in the theory of several complex variables and complex manifolds is a complex submanifold of the vector space of n complex dimensions. The name is for Karl Stein.- Definition :...

 of dimension , then in fact is homotopy equivalent to a CW complex

CW complex

In topology, a CW complex is a type of topological space introduced by J. H. C. Whitehead to meet the needs of homotopy theory. This class of spaces is broader and has some better categorical properties than simplicial complexes, but still retains a combinatorial naturethat allows for...

 of real dimension at most n. In other words
has only half as much topology.

Consequently, if is a closed connected complex submanifold of complex dimension . Then has the homotopy type of a complex of real dimension .
Therefore
, for

and
, for .

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