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In mathematics, the Andreotti–Frankel theorem, introduced by , states that if V is a
smooth Smooth may refer to: Mathematics * Smooth function, a function that is infinitely differentiable; used in calculus and topology * Smooth manifold, a differentiable manifold for which all the transition maps are smooth functions * Smooth algebrai ...
, 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 ideal. ...
of
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 a ...
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 Ste ...
of dimension n, then V admits a
Morse function In mathematics, specifically in differential topology, Morse theory enables one to analyze the topology of a manifold by studying differentiable functions on that manifold. According to the basic insights of Marston Morse, a typical differentiab ...
with critical points of index at most ''n'', and so V is
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 deforma ...
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 cl ...
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