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.
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 H0,2(X) ⊕ H1,1(X) ⊕ H2,0(X), and it can be proven that the image of c1 lies in H1,1(X).
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[2] worked on projective surfaces and used normal functions, which were introduced by Poincaré.
Fix an embedding of X in PN, and choose a pencil of curves Ct on X.
Then the divisor p1(t) + ... + pd(t) − dp0 is a divisor of degree zero, and consequently it determines a class νΓ(t) in the Jacobian JCt 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 H2(X, Z) and that the class of νΓ is the fundamental class of Γ.
Because X is a complex manifold, it admits an exponential sheaf sequence[3] Taking sheaf cohomology of this exact sequence gives maps The group Pic X of line bundles on X is isomorphic to
The first Chern class map is c1 by definition, so it suffices to show that