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.
The Lefschetz theorem refers to any of the following statements:[1][2] Using a long exact sequence, one can show that each of these statements is equivalent to a vanishing theorem for certain relative topological invariants.
Because a generic hyperplane section is smooth, all but a finite number of
-plane and making an additional finite number of slits, the resulting family of hyperplane sections is topologically trivial.
, therefore, can be understood if one understands how hyperplane sections are identified across the slits and at the singular points.
At the singular points, the Morse lemma implies that there is a choice of coordinate system for
This coordinate system can be used to prove the theorem directly.
[4] Aldo Andreotti and Theodore Frankel[5] recognized that Lefschetz's theorem could be recast using Morse theory.
The basic tool in this approach is the Andreotti–Frankel theorem, which states that a complex affine variety of complex dimension
) has the homotopy type of a CW-complex of (real) dimension
The long exact sequence of relative homology then gives the theorem.
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.
An application of Morse theory to this section implies that
Kunihiko Kodaira and Donald C. Spencer found that under certain restrictions, it is possible to prove a Lefschetz-type theorem for the Hodge groups
Combining this proof with the universal coefficient theorem nearly yields the usual Lefschetz theorem for cohomology with coefficients in any field of characteristic zero.
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.
[10] The motivation behind Artin and Grothendieck's proof for constructible sheaves was to give a proof that could be adapted to the setting of étale and
Up to some restrictions on the constructible sheaf, the Lefschetz theorem remains true for constructible sheaves in positive characteristic.
In this setting, the theorem holds for highly singular spaces.
A Lefschetz-type theorem also holds for Picard groups.
-fold product with the cohomology class of a hyperplane gives an isomorphism between
This is the hard Lefschetz theorem, christened in French by Grothendieck more colloquially as the Théorème de Lefschetz vache.
[12][13] 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.
-adic cohomology of smooth projective varieties over algebraically closed fields of positive characteristic by Pierre Deligne (1980).