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.

Let X be a compact Kähler manifold. The first Chern class c₁ gives a map from holomorphic line bundles to H²(X, Z). By Hodge theory, the de Rham cohomology group H²(X, C) decomposes as a direct sum H^0,2(X) ⊕ H^1,1(X) ⊕ H^2,0(X), and it can be proven that the image of c₁ lies in H^1,1(X). The theorem says that the map to H²(X, Z) ∩ H^1,1(X) 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 c₁(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.

Lefschetz's original proof 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 ${\displaystyle {\mathcal {J}}}$, 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.

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