In mathematics, the Andreotti–Frankel theorem, introduced by , states that if

V

is a smooth, complex

affine variety In algebraic geometry, an affine variety, or affine algebraic variety, over an algebraically closed field is the zero-locus in the affine space of some finite family of polynomials of variables with coefficients in that generate a prime ide ...

complex dimension In mathematics, complex dimension usually refers to the dimension of a complex manifold or a complex algebraic variety. These are spaces in which the local neighborhoods of points (or of non-singular points in the case of a variety) are modeled on ...

n

or, more generally, if

V

is any

Stein manifold In mathematics, in the theory of several complex variables and complex manifolds, a Stein manifold is a complex submanifold of the vector space of ''n'' complex dimensions. They were introduced by and named after . A Stein space is similar to a St ...

of dimension

n

, then

V

admits a Morse function with critical points of index at most ''n'', and so

V

homotopy equivalent In topology, a branch of mathematics, two continuous functions from one topological space to another are called homotopic (from grc, ὁμός "same, similar" and "place") if one can be "continuously deformed" into the other, such a defo ...

to a

CW complex A CW complex (also called cellular complex or cell complex) is a kind of a topological space that is particularly important in algebraic topology. It was introduced by J. H. C. Whitehead (open access) to meet the needs of homotopy theory. This cla ...

of real dimension at most ''n''. Consequently, if

V \subseteq \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; \Z)=0,\texti>n

and :

H_i(V; \Z)=0,\texti>n.

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

n

References

* * Chapter 7. Complex manifolds Theorems in homotopy theory {{topology-stub