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 divisors 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. There is a cycle class map that takes a divisor class to a cohomology class. In this case, it is the first Chern class c1 from linear equivalence classes of divisors 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 proved that the image of the cycle class map lies in H1,1(X). The theorem says that the map to H2(X, Z) ∩ H1,1(X) is surjective.

Proof using normal functions

Lefschetz's original proof[2] worked on projective surfaces and used normal functions, which were introduced by Poincaré. Suppose that Ct is a pencil of curves on X. Each of these curves has a Jacobian variety JCt (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 PN, and choose a pencil of curves Ct on X. For a fixed curve Γ on X, the intersection of Γ and Ct is a divisor p1(t) + ... + pd(t) on Ct, where d is the degree of X. Fix a base point p0 of the pencil. 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. Furthermore, he proved that a class in H2(X, Z) is the class of a normal function if and only if it lies in H1,1. 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 of linear equivalence classes of divisors is isomorphic to the group Pic X of line bundles on X, and this group 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 H0,2(X). It follows that it is zero on H2(X, Z) ∩ H1,1(X), and consequently that the cycle class map is surjective.[4]

References

1. ^ Griffiths & Harris 1994, p. 163
2. ^ Lefschetz 1924
3. ^ Griffiths & Harris 1994, p. 37
4. ^ Griffiths & Harris 1994, pp. 163–164