Lefschetz theorem on (1,1)-classes

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[2] 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.

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 p₁(t) + ... + p_d(t) on C_t, where d is the degree of X. Fix a base point p₀ of the pencil. Then the divisor p₁(t) + ... + p_d(t) − dp₀ 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 H²(X, Z) and that the class of ν_Γ is the fundamental class of Γ. Furthermore, he proved that a class in H²(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.

Because X is a complex manifold, it admits an exponential sheaf sequence[3]

${\displaystyle 0\to {\underline {\mathbf {Z} }}{\stackrel {2\pi i}{\longrightarrow }}{\mathcal {O}}_{X}{\stackrel {\operatorname {exp} }{\longrightarrow }}{\mathcal {O}}_{X}^{\times }\to 0.}$

Taking sheaf cohomology of this exact sequence gives maps

${\displaystyle H^{1}(X,{\mathcal {O}}_{X}^{\times }){\stackrel {c_{1}}{\to }}H^{2}(X,\mathbf {Z} ){\stackrel {i_{*}}{\to }}H^{2}(X,{\mathcal {O}}_{X}).}$

The group Pic X of line bundles on X is isomorphic to ${\displaystyle H^{1}(X,{\mathcal {O}}_{X}^{\times })}$. The first Chern class map is c₁ by definition, so it suffices to show that i_* is zero.

Because X is Kähler, Hodge theory implies that ${\displaystyle H^{2}(X,{\mathcal {O}}_{X})\cong H^{0,2}(X)}$. However, i_* factors through the map from H²(X, Z) to H²(X, C), and on H²(X, C), i_* is the restriction of the projection onto H^0,2(X). It follows that it is zero on H²(X, Z) ∩ H^1,1(X), and consequently that the cycle class map is surjective.[4]

• Griffiths, Phillip; Harris, Joseph (1994), Principles of algebraic geometry, Wiley Classics Library, New York: John Wiley & Sons, doi:10.1002/9781118032527, ISBN 978-0-471-05059-9, MR 1288523
• Lefschetz, Solomon (1924), L'Analysis situs et la géométrie algébrique, Collection de Monographies publiée sous la Direction de M. Émile Borel (in French), Paris: Gauthier-Villars Reprinted in Lefschetz, Solomon (1971), Selected papers, New York: Chelsea Publishing Co., ISBN 978-0-8284-0234-7, MR 0299447