mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...

, the theorem of the cube is a condition for a

line bundle In mathematics, a line bundle expresses the concept of a line that varies from point to point of a space. For example, a curve in the plane having a tangent line at each point determines a varying line: the ''tangent bundle'' is a way of organisin ...

over a product of three complete varieties to be trivial. It was a principle discovered, in the context of

linear equivalence In algebraic geometry, divisors are a generalization of codimension-1 subvarieties of algebraic varieties. Two different generalizations are in common use, Cartier divisors and Weil divisors (named for Pierre Cartier (mathematician), Pierre Cartier ...

, by the Italian school of algebraic geometry. The final version of the theorem of the cube was first published by , who credited it to

André Weil André Weil (; ; 6 May 1906 – 6 August 1998) was a French mathematician, known for his foundational work in number theory and algebraic geometry. He was a founding member and the ''de facto'' early leader of the mathematical Bourbaki group. Th ...

. A discussion of the history has been given by . A treatment by means of

sheaf cohomology In mathematics, sheaf cohomology is the application of homological algebra to analyze the global sections of a sheaf on a topological space. Broadly speaking, sheaf cohomology describes the obstructions to solving a geometric problem globally when i ...

, and description in terms of the Picard functor, was given by .

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'' × , ''U''× × ''W'', and × ''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 ''O''_''X'', as an ''O''_''X''-module. If the base ''X'' is a

complex manifold In differential geometry and complex geometry, a complex manifold is a manifold with an atlas of charts to the open unit disc in \mathbb^n, such that the transition maps are holomorphic. The term complex manifold is variously used to mean a com ...

, 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.Alexander Polishchuk, ''Abelian Varieties, Theta Functions and the Fourier Transform'' (2003), p. 122.

Theorem of the square

The theorem of the square is a corollary (also due to Weil) applying to an

abelian variety In mathematics, particularly in algebraic geometry, complex analysis and algebraic number theory, an abelian variety is a projective algebraic variety that is also an algebraic group, i.e., has a group law that can be defined by regular func ...

''A''. One version of it states that the function φ_''L'' taking ''x''∈''A'' to ''T'L''⊗''L''⁻¹ is a group homomorphism from ''A'' to ''Pic''(''A'') (where ''T'' is translation by ''x'' on line bundles).

References

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Notes

{{Reflist Abelian varieties Algebraic varieties Theorems in geometry