Lefschetz theorem on (1,1)-classes [SEP] named after
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" gives a map from holomorphic line bundles to . By Hodge
of degree formula_13. This proves the Hodge conjecture when formula_9 has dimension at most three.
The Lefschetz theorem on (1,1)-classes also implies that if all Hodge classes are generated by the Hodge classes of divisors, then the Hodge conjecture is true:
Known cases of the Hodge conjecture Hypersurfaces.
By the strong and weak Lefschetz theorem, the only non-trivial part of the Hodge conjecture for hypersurfaces is the degree "m" part (i.e., the middle cohomology) of a 2"m"-dimensional hypersurface formula_19. If the degree "