Lefschetz theorem

Lefschetz' fixed-point theorem, or the Lefschetz–Hopf theorem, is a theorem that makes it possible to express the number of fixed points of a continuous mapping in terms of its Lefschetz number. Thus, if a continuous mapping of a finite CW-complex (cf. also Cellular space) has no fixed points, then its Lefschetz number is equal to zero. A special case of this assertion is Brouwer's fixed-point theorem (cf. Brouwer theorem).

References

[a1]	M.J. Greenberg, J.R. Harper, "Algebraic topology, a first course" , Benjamin/Cummings (1981)

Lefschetz' hyperplane-section theorem, or the weak Lefschetz theorem: Let be an algebraic subvariety (cf. Algebraic variety) of complex dimension in the complex projective space , let be a hyperplane passing through all singular points of (if any) and let be a hyperplane section of ; then the relative homology groups (cf. Homology group) vanish for . This implies that the natural homomorphism

is an isomorphism for and is surjective for (see ).

Using universal coefficient formulas (cf. Künneth formula) one obtains corresponding assertions for arbitrary cohomology groups. In every case, for cohomology with coefficients in the field of rational numbers the dual assertions hold: The homomorphism of cohomology spaces

induced by the imbedding is an isomorphism for and is injective for (see ).

An analogous assertion is true for homotopy groups: for . In particular, the canonical homomorphism is an isomorphism for and is surjective for (the Lefschetz theorem on the fundamental group). There is a generalization of this theorem to the case of an arbitrary algebraically closed field (see ), and also to the case when is a normal complete intersection of (see ).

The hard Lefschetz theorem is a theorem about the existence of a Lefschetz decomposition of the cohomology of a complex Kähler manifold into primitive components.

Let be a compact Kähler manifold of dimension with Kähler form , let

be the cohomology class of type corresponding to under the de Rham isomorphism (cf. de Rham cohomology; if is a projective algebraic variety over with the natural Hodge metric, then is the cohomology class dual to the homology class of a hyperplane section) and let

be the linear operator defined by multiplication by , that is,

One has the isomorphism (see )

for any . The kernel of the operator

is denoted by and is called the primitive part of the -cohomology of the variety . The elements of are called primitive cohomology classes, and the cycles corresponding to them are called primitive cycles. The hard Lefschetz theorem establishes the following decomposition of the cohomology into the direct sum of primitives (called the Lefschetz decomposition):

for all . The mappings

are imbeddings. The Lefschetz decomposition commutes with the Hodge decomposition (cf. Hodge conjecture)

(see ). In particular, the primitive part of is defined and

The hard Lefschetz theorem and the Lefschetz decomposition have analogues in abstract algebraic geometry for -adic and crystalline cohomology (see , ).

The Lefschetz theorem on cohomology of type is a theorem about the correspondence between the two-dimensional algebraic cohomology classes of a complex algebraic variety and the cohomology classes of type .

Let be a non-singular projective algebraic variety over the field . An element is said to be algebraic if the cohomology class dual to it (in the sense of Poincaré) is determined by a certain divisor. The Lefschetz theorem on cohomology of type asserts that a class is algebraic if and only if

where is the Hodge component of type of the two-dimensional complex cohomology space , and the mapping is induced by the natural imbedding (see [1], and also [6], [12]). For algebraic cohomology classes in dimensions greater than 2, see Hodge conjecture.

For an arbitrary complex-analytic manifold there is an analogous characterization of elements of the group that are Chern classes of complex line bundles over (see [11]).

References

[1]	S. Lefschetz, "L'analysis situs et la géométrie algébrique" , Gauthier-Villars (1950)
[2]	S. Lefschetz, "On certain numerical invariants of algebraic varieties with applications to Abelian varieties" Trans. Amer. Math. Soc. , 22 (1921) pp. 327–482
[3]	S. Lefschetz, "On the fixed point formula" Ann. of Math. (2) , 38 (1937) pp. 819–822
[4]	P. Berthelot, "Cohomologie cristalline des schémas de caractéristique " , Springer (1974)
[5]	P. Deligne (ed.) N.M. Katz (ed.) , Groupes de monodromie en géométrie algébrique. SGA 7.II , Lect. notes in math. , 340 , Springer (1973)
[6]	P.A. Griffiths, J.E. Harris, "Principles of algebraic geometry" , Wiley (Interscience) (1978)
[7]	A. Grothendieck, "Cohomologie locale des faisceaux cohérents et théorèmes de Lefschetz locaux et globaux" , SGA 2 , North-Holland & Masson (1968)
[8]	R. Hartshorne, "Ample subvarieties of algebraic varieties" , Springer (1970)
[9]	D. Mumford, "Abelian varieties" , Oxford Univ. Press (1974)
[10]	J.W. Milnor, "Morse theory" , Princeton Univ. Press (1963)
[11]	R.O. Wells jr., "Differential analysis on complex manifolds" , Springer (1980)
[12]	S.S. Chern, "Complex manifolds without potential theory" , Springer (1979)
[13]	A. Weil, "Introduction à l'Aeetude des variétés kahlériennes" , Hermann (1958)
[14]	P. Deligne, "La conjecture de Weil" Publ. Math. IHES , 43 (1974) pp. 273–307

V.A. Iskovskikh

Comments

For a modern treatment of the classical Lefschetz hyperplane-section theorems see [a1].

Weak and hard (strong) Lefschetz theorems also hold in étale cohomology [a4] and in intersection homology [a5], [a6]. For the proof of the hard Lefschetz theorem in -adic cohomology, see [a2].

References

[a1]	K. Lamotke, "The topology of complex projective varieties after S. Lefschetz" Topology , 20 (1981) pp. 15–51
[a2]	P. Deligne, "La conjecture de Weil II" Publ. Math. IHES , 52 (1980) pp. 137–252
[a3]	M.J. Greenberg, J.R. Harper, "Algebraic topology, a first course" , Benjamin/Cummings (1981)
[a4]	J.S. Milne, "Etale cohomology" , Princeton Univ. Press (1980)
[a5]	M. Goresky, "Stratified Morse theory" , Springer (1988)
[a6]	A. Beilinson, J. Bernstein, P. Deligne, "Faisceaux pervers" Astérisque , 100 (1982)