Lefschetz Hyperplane Theorem
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In mathematics, specifically in algebraic geometry and
algebraic topology Algebraic topology is a branch of mathematics that uses tools from abstract algebra to study topological spaces. The basic goal is to find algebraic invariants that classify topological spaces up to homeomorphism, though usually most classify ...
, the Lefschetz hyperplane theorem is a precise statement of certain relations between the shape of an
algebraic variety Algebraic varieties are the central objects of study in algebraic geometry, a sub-field of mathematics. Classically, an algebraic variety is defined as the set of solutions of a system of polynomial equations over the real or complex numbers. ...
and the shape of its subvarieties. More precisely, the theorem says that for a variety ''X'' embedded in projective space and a
hyperplane section In mathematics, a hyperplane section of a subset ''X'' of projective space P''n'' is the intersection of ''X'' with some hyperplane ''H''. In other words, we look at the subset ''X'H'' of those elements ''x'' of ''X'' that satisfy the single line ...
''Y'', the
homology Homology may refer to: Sciences Biology *Homology (biology), any characteristic of biological organisms that is derived from a common ancestor * Sequence homology, biological homology between DNA, RNA, or protein sequences *Homologous chrom ...
,
cohomology In mathematics, specifically in homology theory and algebraic topology, cohomology is a general term for a sequence of abelian groups, usually one associated with a topological space, often defined from a cochain complex. Cohomology can be viewe ...
, and homotopy groups of ''X'' determine those of ''Y''. A result of this kind was first stated by
Solomon Lefschetz Solomon Lefschetz (russian: Соломо́н Ле́фшец; 3 September 1884 – 5 October 1972) was an American mathematician who did fundamental work on algebraic topology, its applications to algebraic geometry, and the theory of non-linear o ...
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 In mathematics, especially algebraic geometry, the decomposition theorem of Beilinson, Bernstein and Deligne or BBD decomposition theorem is a set of results concerning the cohomology of algebraic varieties. It was originally conjectured by Gelfand ...
.


The Lefschetz hyperplane theorem for complex projective varieties

Let ''X'' be an ''n''-dimensional complex projective algebraic variety in CP''N'', 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 ''H''''k''(''Y'', Z) → ''H''''k''(''X'', Z) in singular homology is an isomorphism for ''k'' < ''n'' − 1 and is surjective for ''k'' = ''n'' − 1. # The natural map ''H''''k''(''X'', Z) → ''H''''k''(''Y'', Z) in singular cohomology is an isomorphism for ''k'' < ''n'' − 1 and is injective for ''k'' = ''n'' − 1. # The natural map π''k''(''Y'', Z) → π''k''(''X'', Z) is an isomorphism for ''k'' < ''n'' − 1 and is surjective for ''k'' = ''n'' − 1. Using a
long exact sequence An exact sequence is a sequence of morphisms between objects (for example, groups, rings, modules, and, more generally, objects of an abelian category) such that the image of one morphism equals the kernel of the next. Definition In the context ...
, 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 ''H''''k''(''X'', ''Y'', Z) are zero for k \leq n-1. # The relative singular cohomology groups ''H''''k''(''X'', ''Y'', Z) are zero for k \leq n-1. # The relative homotopy groups π''k''(''X'', ''Y'') are zero for k \leq n-1.


Lefschetz's proof

Solomon Lefschetz Solomon Lefschetz (russian: Соломо́н Ле́фшец; 3 September 1884 – 5 October 1972) was an American mathematician who did fundamental work on algebraic topology, its applications to algebraic geometry, and the theory of non-linear o ...
used his idea of a
Lefschetz pencil In mathematics, a Lefschetz pencil is a construction in algebraic geometry considered by Solomon Lefschetz, used to analyse the algebraic topology of an algebraic variety ''V''. Description A ''pencil'' is a particular kind of linear system of ...
to prove the theorem. Rather than considering the hyperplane section ''Y'' alone, he put it into a family of hyperplane sections ''Y''''t'', where ''Y'' = ''Y''0. Because a generic hyperplane section is smooth, all but a finite number of ''Y''''t'' 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. That is, it is a product of a generic ''Y''''t'' with an open subset of the ''t''-plane. ''X'', 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 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 differenti ...
implies that there is a choice of coordinate system for ''X'' of a particularly simple form. This coordinate system can be used to prove the theorem directly.


Andreotti and Frankel's proof

Aldo Andreotti Aldo Andreotti (15 March 1924 – 21 February 1980) was an Italian mathematician who worked on algebraic geometry, on the theory of functions of several complex variables and on partial differential operators. Notably he proved the Andreotti– ...
and
Theodore Frankel Theodore Frankel (June 17, 1929 – August 5, 2017) was a mathematician who introduced the Andreotti–Frankel theorem and the Frankel conjecture. Frankel received his Ph.D. from the University of California, Berkeley in 1955. His doctoral adv ...
recognized that Lefschetz's theorem could be recast using
Morse theory 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 ...
. Here the parameter ''t'' plays the role of a Morse function. The basic tool in this approach is the
Andreotti–Frankel theorem In mathematics, the Andreotti–Frankel theorem, introduced by , states that if V is a smooth algebraic variety, smooth, complex affine variety of complex dimension n or, more generally, if V is any Stein manifold of dimension n, then V admits a ...
, which states that a 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 ''n'' (and thus real dimension 2''n'') has the homotopy type of 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 clas ...
of (real) dimension ''n''. This implies that the
relative homology In algebraic topology, a branch of mathematics, the (singular) homology of a topological space relative to a subspace is a construction in singular homology, for pairs of spaces. The relative homology is useful and important in several ways. Intui ...
groups of ''Y'' in ''X'' are trivial in degree less than ''n''. 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 Raoul Bott (September 24, 1923 – December 20, 2005) was a Hungarian-American mathematician known for numerous basic contributions to geometry in its broad sense. He is best known for his Bott periodicity theorem, the Morse–Bott functions whi ...
in 1959. Thom and Bott interpret ''Y'' as the vanishing locus in ''X'' of a section of a line bundle. An application of Morse theory to this section implies that ''X'' can be constructed from ''Y'' by adjoining cells of dimension ''n'' or more. From this, it follows that the relative homology and homotopy groups of ''Y'' in ''X'' are concentrated in degrees ''n'' and higher, which yields the theorem.


Kodaira and Spencer's proof for Hodge groups

Kunihiko Kodaira was a Japanese mathematician known for distinguished work in algebraic geometry and the theory of complex manifolds, and as the founder of the Japanese school of algebraic geometers. He was awarded a Fields Medal in 1954, being the first Japanese ...
and Donald C. Spencer found that under certain restrictions, it is possible to prove a Lefschetz-type theorem for the Hodge groups ''H''''p'',''q''. Specifically, assume that ''Y'' is smooth and that the line bundle \mathcal_X(Y) is ample. Then the restriction map ''H''''p'',''q''(''X'') → ''H''''p'',''q''(''Y'') is an isomorphism if and is injective if ''p'' + ''q'' = ''n'' − 1. By Hodge theory, these cohomology groups are equal to the sheaf cohomology groups H^q(X, \textstyle\bigwedge^p\Omega_X) and H^q(Y, \textstyle\bigwedge^p\Omega_Y). Therefore, the theorem follows from applying the
Akizuki–Nakano vanishing theorem In mathematics, specifically in the study of vector bundles over complex Kähler manifolds, the Nakano vanishing theorem, sometimes called the Akizuki–Nakano vanishing theorem, generalizes the Kodaira vanishing theorem. Given a compact complex m ...
to H^q(X, \textstyle\bigwedge^p\Omega_X, _Y) and using a long exact sequence. Combining this proof with the
universal coefficient theorem In algebraic topology, universal coefficient theorems establish relationships between homology groups (or cohomology groups) with different coefficients. For instance, for every topological space , its ''integral homology groups'': : completely ...
nearly yields the usual Lefschetz theorem for cohomology with coefficients in any field of characteristic zero. It is, however, slightly weaker because of the additional assumptions on ''Y''.


Artin and Grothendieck's proof for constructible sheaves

Michael Artin Michael Artin (; born 28 June 1934) is a German-American mathematician and a professor emeritus in the Massachusetts Institute of Technology mathematics department, known for his contributions to algebraic geometry.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 In mathematics, a constructible sheaf is a sheaf of abelian groups over some topological space ''X'', such that ''X'' is the union of a finite number of locally closed subsets on each of which the sheaf is a locally constant sheaf. It has its origi ...
. They prove that for a constructible sheaf ''F'' on an affine variety ''U'', the cohomology groups H^k(U,F) vanish whenever k>n.


The Lefschetz theorem in other cohomology theories

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 \ell-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 topology, a branch of mathematics, intersection homology is an analogue of singular homology especially well-suited for the study of singular spaces, discovered by Mark Goresky and Robert MacPherson in the fall of 1974 and developed by them ov ...
. In this setting, the theorem holds for highly singular spaces. A Lefschetz-type theorem also holds for
Picard group In mathematics, the Picard group of a ringed space ''X'', denoted by Pic(''X''), is the group of isomorphism classes of invertible sheaves (or line bundles) on ''X'', with the group operation being tensor product. This construction is a global ve ...
s.


Hard Lefschetz theorem

Let ''X'' be a ''n''-dimensional non-singular complex projective variety in \mathbb^N. Then in the cohomology ring of ''X'', the ''k''-fold product with the cohomology class of a hyperplane gives an isomorphism between H^(X) and H^(X). 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 In mathematics and especially differential geometry, a Kähler manifold is a manifold with three mutually compatible structures: a complex structure, a Riemannian structure, and a symplectic structure. The concept was first studied by Jan Arn ...
, 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 surface In complex geometry, a Hopf surface is a compact complex surface obtained as a quotient of the complex vector space (with zero deleted) \Complex^2\setminus \ by a free action of a discrete group. If this group is the integers the Hopf surface is c ...
s 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

* * * * * * * Reprinted in * * * {{Citation , last=Voisin , first=Claire , authorlink=Claire Voisin, title=Hodge theory and complex algebraic geometry. II , publisher=
Cambridge University Press Cambridge University Press is the university press of the University of Cambridge. Granted letters patent by King Henry VIII in 1534, it is the oldest university press in the world. It is also the King's Printer. Cambridge University Pre ...
, series=Cambridge Studies in Advanced Mathematics , isbn=978-0-521-80283-3 , mr=1997577 , year=2003 , volume=77 , doi=10.1017/CBO9780511615177 Topological methods of algebraic geometry Morse theory Theorems in algebraic geometry Theorems in algebraic topology