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
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. A discussion of the history has been given by . A treatment by means of
sheaf cohomology, 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 ...
, 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 coherent sheaf ''S'' on a ringed space ''X'', for which there is an inverse ''T'' with respect to tensor product of ''O'X''-modules. It is the equivalent in algebraic geometry of the topological notion of ...
''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''
''X'', as an ''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 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 (or, in 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''.
Definition
To an abelian variety ''A'' over a field ''k'', one associates a dual abelian variety ''A''v (over the same field), which i ...
.
[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 ''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