Theorem of the cube

In mathematics, the theorem of the cube is a condition for a line bundle over a product of three complete varieties to be trivial.

It was a principle discovered, in the context of linear equivalence, by the Italian school of algebraic geometry.

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

[1] The theorem of the square (Lang 1959) (Mumford 2008, p.59) is a corollary (also due to Weil) applying to an abelian variety A.

One version of it states that the function φL taking x∈A to T*xL⊗L−1 is a group homomorphism from A to Pic(A) (where T*x is translation by x on line bundles).