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.

The Lefschetz hyperplane theorem for complex projective varieties

is smooth. The Lefschetz theorem refers to any of the following statements:

1. in singular homology.

2. .

3. .

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:

1. .

2. .

3. .

Lefschetz's proof

Solomon Lefschetz used his idea of a Lefschetz pencil, therefore, can be understood if one understands how hyperplane sections are identified across the slits and at the singular points. Away from the singular points, the identification can be described inductively. At the singular points, the Morse lemma of a particularly simple form. This coordinate system can be used to prove the theorem directly.

Andreotti and Frankel's proof

Aldo Andreotti and Theodore Frankel recognized that Lefschetz's theorem could be recast using Morse theory. plays the role of a Morse function. The basic tool in this approach is the Andreotti–Frankel theorem, which states that a complex affine variety) has the homotopy type of a CW-complex. This implies that the relative homology. The long exact sequence of relative homology then gives the theorem.

Thom's and Bott's proofs

Neither Lefschetz's proof nor Andreotti and Frankel's proof directly imply the Lefschetz hyperplane theorem for homotopy groups. An approach that does was found by René Thom no later than 1957 and was simplified and published by Raoul Bott in 1959. and higher, which yields the theorem.

Kodaira and Spencer's proof for Hodge groups

Kunihiko Kodaira and Donald C. Spencer. By Hodge theory, these cohomology groups are equal to the sheaf cohomology. Therefore, the theorem follows from applying the Akizuki–Nakano vanishing theorem and using a long exact sequence.

Combining this proof with the universal coefficient theorem.

Artin and Grothendieck's proof for constructible sheaves

Michael Artin and Alexander Grothendieck found a generalization of the Lefschetz hyperplane theorem to the case where the coefficients of the cohomology lie not in a field but instead in a constructible sheaf.

The Lefschetz theorem in other cohomology theories

-adic cohomology. Up to some restrictions on the constructible sheaf, the Lefschetz theorem remains true for constructible sheaves in positive characteristic.

The theorem can also be generalized to intersection homology. In this setting, the theorem holds for highly singular spaces.

A Lefschetz-type theorem also holds for Picard groups.

Hard Lefschetz theorem

. Then in the cohomology ring-fold product with the cohomology class.

This is the hard Lefschetz theorem, christened in French by Grothendieck more colloquially as the Théorème de Lefschetz vache. It immediately implies the injectivity part of the Lefschetz hyperplane theorem.

The hard Lefschetz theorem in fact holds for any compact Kähler manifold, with the isomorphism in de Rham cohomology given by multiplication by a power of the class of the Kähler form. It can fail for non-Kähler manifolds: for example, Hopf surfaces have vanishing second cohomology groups, so there is no analogue of the second cohomology class of a hyperplane section.

The hard Lefschetz theorem was proven for \ell-adic cohomology of smooth projective varieties over algebraically closed fields of positive characteristic by .

References

Bibliography

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