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Concept
Lefschetz hyperplane theorem
Summary
In mathematics, specifically in algebraic geometry and algebraic topology, the Lefschetz hyperplane theorem is a precise statement of certain relations between the shape of an algebraic variety and the shape of its subvarieties. More precisely, the theorem says that for a variety X embedded in projective space and a hyperplane section Y, the homology, cohomology, and homotopy groups of X determine those of Y. A result of this kind was first stated by Solomon Lefschetz for homology groups of complex algebraic varieties. Similar results have since been found for homotopy groups, in positive characteristic, and in other homology and cohomology theories.
A far-reaching generalization of the hard Lefschetz theorem is given by the decomposition theorem.
Let X be an n-dimensional complex projective algebraic variety in CPN, and let Y be a hyperplane section of X such that U = X ∖ Y is smooth. The Lefschetz theorem refers to any of the following statements:
The natural map Hk(Y, Z) → Hk(X, Z) in singular homology is an isomorphism for k < n − 1 and is surjective for k = n − 1.
The natural map Hk(X, Z) → Hk(Y, Z) in singular cohomology is an isomorphism for k < n − 1 and is injective for k = n − 1.
The natural map πk(Y, Z) → πk(X, Z) is an isomorphism for k < n − 1 and is surjective for k = n − 1.
Using a long exact sequence, one can show that each of these statements is equivalent to a vanishing theorem for certain relative topological invariants. In order, these are:
The relative singular homology groups Hk(X, Y, Z) are zero for .
The relative singular cohomology groups Hk(X, Y, Z) are zero for .
The relative homotopy groups πk(X, Y) are zero for .
Solomon Lefschetz used his idea of a Lefschetz pencil to prove the theorem. Rather than considering the hyperplane section Y alone, he put it into a family of hyperplane sections Yt, where Y = Y0. Because a generic hyperplane section is smooth, all but a finite number of Yt are smooth varieties. After removing these points from the t-plane and making an additional finite number of slits, the resulting family of hyperplane sections is topologically trivial.
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