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.^[1]

Statement of the theorem

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.

Proof using normal functions

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 $\mathcal{J}$ , 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.

Proof using sheaf cohomology

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

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 c₁ by definition, so it suffices to show that i_* is zero.

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 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]

References

Bibliography

Griffiths, Phillip; Harris, Joseph (1994), Principles of algebraic geometry, Wiley Classics Library, New York: John Wiley & Sons, 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