mathematics Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...

, 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 organis ...

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 and André Weil by David Mumfo ...

, by the

Italian school of algebraic geometry In relation to the history of mathematics, the Italian school of algebraic geometry refers to mathematicians and their work in birational geometry, particularly on algebraic surfaces, centered around Rome roughly from 1885 to 1935. There were 30 ...

. 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 one of the most influential mathematicians of the twentieth century. His influence is du ...

. 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 (holes) to solving a geometric problem glob ...

, and description in terms of the

Picard functor In mathematics, the Picard group of a ringed space ''X'', denoted by Pic(''X''), is the group of isomorphism classes of invertible sheaves (or line bundles) on ''X'', with the group operation being tensor product. This construction is a global ver ...

, 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 In mathematics, an invertible sheaf is a sheaf on a ringed space that has an inverse with respect to tensor product of sheaves of modules. It is the equivalent in algebraic geometry of the topological notion of a line bundle. Due to their intera ...

''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 In mathematics, a ringed space is a family of (commutative) rings parametrized by open subsets of a topological space together with ring homomorphisms that play roles of restrictions. Precisely, it is a topological space equipped with a sheaf of ...

''X'', an invertible sheaf ''L'' is ''trivial'' if isomorphic to ''O'', as an ''O''-module. If the base ''X'' is a

complex manifold In differential geometry and complex geometry, a complex manifold is a manifold with a ''complex structure'', that is an atlas (topology), atlas of chart (topology), charts to the open unit disc in the complex coordinate space \mathbb^n, such th ...

, then an invertible sheaf is (the sheaf of sections of) a

holomorphic line bundle In mathematics, a holomorphic vector bundle is a complex vector bundle over a complex manifold such that the total space is a complex manifold and the projection map is holomorphic. Fundamental examples are the holomorphic tangent bundle of a ...

, and trivial means holomorphically equivalent to a

trivial bundle In mathematics, and particularly topology, a fiber bundle ( ''Commonwealth English'': fibre bundle) is a space that is a product space, but may have a different topological structure. Specifically, the similarity between a space E and a p ...

, 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 In mathematics, a dual abelian variety can be defined from an abelian variety ''A'', defined over a field ''k''. A 1-dimensional abelian variety is an elliptic curve, and every elliptic curve is isomorphic to its dual, but this fails for higher- ...

.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 smooth Algebraic variety#Projective variety, projective algebraic variety that is also an algebraic group, i.e., has a group ...

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

References

* * *

Notes

{{Reflist Abelian varieties Algebraic varieties

Cube A cube or regular hexahedron is a three-dimensional space, three-dimensional solid object in geometry, which is bounded by six congruent square (geometry), square faces, a type of polyhedron. It has twelve congruent edges and eight vertices. It i ...