Lefschetz Theorem on (1,1)- Classes
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High Quality Content by WIKIPEDIA articles! In algebraic geometry, a branch of mathematics, the Lefschetz theorem on (1,1)-classes, named after Solomon Lefschetz, is a classical statement relating divisors 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.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 c1 from linear equivalence classes of divisors to H2(X, Z). By Hodge theory, ...
High Quality Content by WIKIPEDIA articles! In algebraic geometry, a branch of mathematics, the Lefschetz theorem on (1,1)-classes, named after Solomon Lefschetz, is a classical statement relating divisors 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.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 c1 from linear equivalence classes of divisors to H2(X, Z). By Hodge theory, the de Rham cohomology group H2(X, C) decomposes as a direct sum H0,2(X) H1,1(X) H2,0(X), and it can be proved that the image of the cycle class map lies in H1,1(X). The theorem says that the map to H2(X, Z) H1,1(X) is surjective.
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