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Lefschetz theorem on (1,1)-classes
In algebraic geometry, a branch of mathematics, the Lefschetz theorem on (1,1)-classes, named after Solomon Lefschetz, is a classical statement relating holomorphic line bundles on a 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.
== Statement of the theorem ==
Let ''X'' be a compact Kähler manifold. The first Chern class ''c''1 gives a map from holomorphic line bundles to . By Hodge theory, the de Rham cohomology group ''H''2(''X'', C) decomposes as a direct sum , and it can be proven that the image of ''c''1 lies in ''H''1,1(''X''). The theorem says that the map to is surjective.
In the special case where ''X'' is a projective variety, holomorphic line bundles are in bijection with linear equivalences class of divisors, and given a divisor ''D'' on ''X'' with associated line bundle ''O(D)'', the class ''c''1(''O(D)'') is Poincaré dual to the homology class given by ''D''. Thus, this establishes the usual formulation of the Hodge conjecture for divisors in projective varieties.
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