# 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 H$^2$(X, Z). By Hodge theory, the de Rham cohomology group H$^2$(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 c$_1$ lies in H$^{1,1}$(X). The theorem says that the map to H$^2$(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 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.

## Proof using normal functions

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 ${\mathcal {J}}$, 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.
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 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 p$_1$(t) + ... + p$_d$(t) − dp$_0$ is a divisor of degree zero, and consequently it determines a class ν$_Γ$(t) in the Jacobian JC$_t$ for all t. The map from t to ν$_Γ$(t) is a normal function.
Henri Poincaré proved that for a general pencil of curves, all normal functions arose as ν$_Γ$(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 ν$_Γ$ 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
    $0\to {\underline {\mathbf {Z} }}{\stackrel {2\pi i}{\longrightarrow }}{\mathcal {O}}_{X}{\stackrel {\operatorname {exp} }{\longrightarrow }}{\mathcal {O}}_{X}^{\times }\to 0.$
Taking sheaf cohomology of this exact sequence gives maps
    $H^{1}(X,{\mathcal {O}}_{X}^{\times }){\stackrel {c_{1}}{\to }}H^{2}(X,\mathbf {Z} ){\stackrel {i_{*}}{\to }}H^{2}(X,{\mathcal {O}}_{X}).$
The group Pic X of line bundles on X is isomorphic to $H^{1}(X,{\mathcal {O}}_{X}^{\times })$. The first Chern class map is c$_1$ by definition, so it suffices to show that $i_{*}$ is zero on H$^2$(X, Z) ∩ H$^{1,1}$(X).
Because X is Kähler, Hodge theory implies that $H^{2}(X,{\mathcal {O}}_{X})\cong H^{0,2}(X)$. 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 H$^2$(X, Z) ∩ H$^{1,1}$(X), and consequently that the cycle class map is surjective.