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In [[algebraic geometry]], a branch of [[mathematics]], the '''Lefschetz theorem on (1,1)-classes''', named after [[Solomon Lefschetz]], is a classical statement relating [[divisor (algebraic geometry)|divisor]]s on a [[compact space|compact]] [[Kähler manifold]] to classes in its integral [[cohomology]]. It is the only case of the [[Hodge conjecture]] which has been proved for all Kähler manifolds.<ref>{{Harvnb|Griffiths|Harris|1994|p=163}}</ref>

== Statement of the theorem ==
Let ''X'' be a compact Kähler manifold. There is a cycle class map that takes a divisor class to a cohomology class. In this case, it is the first [[Chern class]] ''c''1 from linear equivalence classes of divisors to {{nowrap|''H''2(''X'', '''Z''')}}. By [[Hodge theory]], the [[de Rham cohomology]] group ''H''2(''X'', '''C''') decomposes as a direct sum {{nowrap|''H''0,2(''X'') &oplus; ''H''1,1(''X'') &oplus; ''H''2,0(''X'')}}, and it can be proved that the image of the cycle class map lies in ''H''1,1(''X''). The theorem says that the map to {{nowrap|''H''2(''X'', '''Z''') ∩ ''H''1,1(''X'')}} is surjective.

== Proof using normal functions ==
Lefschetz's original proof<ref>{{Harvnb|Lefschetz|1924}}</ref> worked on projective surfaces and used normal functions, which were introduced by Poincaré. Suppose that ''C''''t'' is a pencil of curves on ''X''. Each of these curves has a [[Jacobian variety]] ''JC''''t'' (if a curve is singular, there is an appropriate generalized Jacobian variety). These can be assembled into a family <math>\mathcal{J}</math>, the Jacobian of the pencil, which comes with a projection map π to the base ''T'' of the pencil. A '''normal function''' is a (holomorphic) section of π.

Fix an embedding of ''X'' in '''P'''''N'', and choose a pencil of curves ''C''''t'' on ''X''. For a fixed curve Γ on ''X'', the intersection of Γ and ''C''''t'' is a divisor {{nowrap|''p''1(''t'') + ... + ''p''''d''(''t'')}} on ''C''''t'', where ''d'' is the degree of ''X''. Fix a base point ''p''0 of the pencil. Then the divisor {{nowrap|''p''1(''t'') + ... + ''p''''d''(''t'') − ''dp''0}} is a divisor of degree zero, and consequently it determines a class νΓ(''t'') in the Jacobian ''JC''''t'' for all ''t''. The map from ''t'' to νΓ(''t'') is a normal function.

[[Henri Poincaré]] proved that for a general pencil of curves, all normal functions arose as νΓ(''t'') for some choice of Γ. Lefschetz proved that any normal function determined a class in ''H''2(''X'', '''Z''') and that the class of νΓ is the fundamental class of Γ. Furthermore, he proved that a class in ''H''2(''X'', '''Z''') is the class of a normal function if and only if it lies in ''H''1,1. Together with Poincaré's existence theorem, this proves the theorem on (1,1)-classes.

== Proof using sheaf cohomology ==
Because ''X'' is a complex manifold, it admits an [[exponential sheaf sequence]]<ref>{{Harvnb|Griffiths|Harris|1994|p=37}}</ref>
:<math>0 \to \underline{\mathbf{Z}} \stackrel{2\pi i}{\longrightarrow} \mathcal{O}_X \stackrel{\operatorname{exp}}{\longrightarrow} \mathcal{O}_X^\times \to 0.</math>
Taking sheaf cohomology of this exact sequence gives maps
:<math>H^1(X, \mathcal{O}_X^\times) \stackrel{c_1}{\to} H^2(X, \mathbf{Z}) \stackrel{i_*}{\to} H^2(X, \mathcal{O}_X).</math>
The group of linear equivalence classes of divisors is isomorphic to the group {{nowrap|Pic ''X''}} of [[line bundle]]s on ''X'', and this group is isomorphic to <math>H^1(X, \mathcal{O}_X^\times)</math>. The first Chern class map is ''c''1 by definition, so it suffices to show that ''i''* is zero.

Because ''X'' is Kähler, Hodge theory implies that <math>H^2(X, \mathcal{O}_X) \cong H^{0,2}(X)</math>. However, ''i''* factors through the map from ''H''2(''X'', '''Z''') to ''H''2(''X'', '''C'''), and on ''H''2(''X'', '''C'''), ''i''* is the restriction of the projection onto ''H''0,2(''X''). It follows that it is zero on {{nowrap|''H''2(''X'', '''Z''') ∩ ''H''1,1(''X'')}}, and consequently that the cycle class map is surjective.<ref>{{Harvnb|Griffiths|Harris|1994|pp=163–164}}</ref>

== References ==
<references/>

== Bibliography ==
* {{Citation | last1=Griffiths | first1=Phillip | author1-link=Phillip Griffiths | last2=Harris | first2=Joseph | author2-link=Joe Harris (mathematician) | title=Principles of algebraic geometry | publisher=[[John Wiley & Sons]] | location=New York | series=Wiley Classics Library | isbn=978-0-471-05059-9 | id={{MathSciNet | id = 1288523}} | year=1994}}
* {{Citation | last1=Lefschetz | first1=Solomon | title=L'Analysis situs et la géométrie algébrique | publisher=Gauthier-Villars | language=French | series=Collection de Monographies publiée sous la Direction de M. Émile Borel | location=Paris | year=1924}} Reprinted in {{Citation | last1=Lefschetz | first1=Solomon | title=Selected papers | publisher=Chelsea Publishing Co. | location=New York | isbn=978-0-8284-0234-7 | id={{MathSciNet | id = 0299447}} | year=1971}}

[[Category:Theorems in algebraic geometry]]

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