In algebraic geometry, the theorem of the cube is a condition for a line bundle over a product of three complete varieties to be trivial. It was first discovered in the context of linear equivalence by the Italian school of algebraic geometry. The modern version of the theorem was first published by Serge Lang, who credited it to André Weil. A treatment by means of sheaf cohomology, and description in terms of the Picard functor, was given by David Mumford.

Statement

The theorem states that for any complete varieties on them, any invertible sheaf is itself trivial.

Special cases

On a ringed space is trivial is a complex manifold, then an invertible sheaf is (the sheaf of sections of) a holomorphic line bundle, and trivial means holomorphically equivalent to a trivial bundle, not just topologically equivalent.

Restatement using biextensions

Weil's result has been restated in terms of biextensions, a concept now generally used in the duality theory of abelian varieties.

Theorem of the square

The theorem of the square is a corollary (also due to Weil) applying to an abelian variety on line bundles).

Notes

References