Theorem of the Cube

The Theorem of the Cube says that the third finite difference of a line bundle on an abelian variety is trivial. On $X^3$, it is the alternating tensor product of the pullbacks along the seven nonempty partial-sum maps.

For a line bundle $L$ on an abelian variety and points $x,y$,

For a line bundle $L$ on an abelian variety $X$,is a homomorphism by the Theorem of the square.

For an abelian variety,is the subgroup of translation-invariant line bundles.

The seesaw theorem determines a line bundle on a product from its restrictions to fibers: if it is trivial on all fibers over one factor and on one transverse section, then it is trivial.