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. The specific result was proved under this name, in the early 1950s, in the course of his fundamental work on abstract algebraic geometry by André Weil; a discussion of the history has been given by Kleiman (2005). A treatment by means of sheaf cohomology, and description in terms of the Picard functor, was given by Mumford (2008).

The theorem states that for any complete varieties U, V and W, and given points u, v and w on them, any invertible sheaf L which has a trivial restriction to each of U× V × {w}, U× {v} × W, and {u} × V × W, is itself trivial. (Mumford p. 55; the result there is slightly stronger, in that one of the varieties need not be complete and can be replaced by a connected scheme.)

Note: On a ringed space X, an invertible sheaf L is trivial if isomorphic to O_X, as O_X-module. If the base X 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.

The theorem of the square (Mumford 2008, p.59) is a corollary applying to an abelian variety A, defining a group homomorphism from A to Pic(A), in terms of the change in L by translation on A.

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

References

• Kleiman, Steven L. (2005), "The Picard scheme", Fundamental algebraic geometry, Math. Surveys Monogr., vol. 123, Providence, R.I.: American Mathematical Society, pp. 235–321, arXiv:math/0504020, MR 2223410
• Mumford, David (2008) [1970], Abelian varieties, Tata Institute of Fundamental Research Studies in Mathematics, vol. 5, Providence, R.I.: American Mathematical Society, ISBN 978-81-85931-86-9, MR 0282985, OCLC 138290