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 final version of the theorem of the cube was first published by Lang (1959), who credited it to 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).
Statement
The theorem states that for any complete varieties U, V and W over an algebraically closed field, 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.)
Special cases
On a ringed space X, an invertible sheaf L is trivial if isomorphic to OX, as an OX-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.
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.[1]
Theorem of the square
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).
References
Kleiman, Steven L. (2005), "The Picard scheme", Fundamental algebraic geometry, Math. Surveys Monogr., 123, Providence, R.I.: American Mathematical Society, pp. 235–321, arXiv:math/0504020, Bibcode:2005math......4020K, MR 2223410
Lang, Serge (1959), Abelian varieties, Interscience Tracts in Pure and Applied Mathematics, 7, New York: Interscience Publishers, Inc., MR 0106225
Mumford, David (2008) [1970], Abelian varieties, Tata Institute of Fundamental Research Studies in Mathematics, 5, Providence, R.I.: American Mathematical Society, ISBN 978-81-85931-86-9, MR 0282985, OCLC 138290
Notes
Alexander Polishchuk, Abelian Varieties, Theta Functions and the Fourier Transform (2003), p. 122.
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