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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 {{harvtxt|Lang|1959}}, who credited it to [[André Weil]]. A discussion of the history has been given by {{harvtxt|Kleiman|2005}}. A treatment by means of [[sheaf cohomology]], and description in terms of the [[Picard functor]], was given
by {{harvtxt|Mumford|2008}}.
==Statement==
The theorem states that for any [[complete variety|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 ''O{{sub|X}}'', as an ''O{{sub|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.
===Restatement using biextensions===
Weil's result has been restated in terms of [[biextension]]s, a concept now generally used in the [[duality theory of abelian varieties]].<ref>Alexander Polishchuk, ''Abelian Varieties, Theta Functions and the Fourier Transform'' (2003), p. 122.</ref>
==Theorem of the square==
The '''theorem of the square''' {{harv|Lang|1959}} {{harv|Mumford|2008|loc= p.59}} is a corollary (also due to Weil) applying to an [[abelian variety]] ''A''. One version of it states that the function φ{{sub|''L''}} taking ''x''∈''A'' to ''T''{{su|p=*|b=x}}''L''⊗''L''{{sup|−1}}  is a group homomorphism from ''A'' to ''Pic''(''A'') (where ''T''{{su|p=*|b=x}} is translation by ''x'' on line bundles).
==References==
*{{Citation | last1=Kleiman | first1=Steven L. | author1-link=Steven Kleiman | title=Fundamental algebraic geometry | arxiv=math/0504020 | publisher=[[American Mathematical Society]] | location=Providence, R.I. | series=Math. Surveys Monogr. | mr=2223410 | year=2005 | volume=123 | chapter=The Picard scheme | pages=235–321| bibcode=2005math......4020K }}
*{{citation|mr=0106225
|last=Lang|first= Serge
|title=Abelian varieties
|series=Interscience Tracts in Pure and Applied Mathematics|volume= 7 |publisher=Interscience Publishers, Inc.|place= New York|year=1959}}
* {{Citation | last1=Mumford | first1=David | author1-link=David Mumford | title=Abelian varieties | origyear=1970 | publisher=[[American Mathematical Society]] | location=Providence, R.I. | series=Tata Institute of Fundamental Research Studies in Mathematics | isbn=978-81-85931-86-9 | oclc=138290 | mr=0282985 | year=2008 | volume=5}}
==Notes==
{{Reflist}}
[[Category:Abelian varieties]]
[[Category:Algebraic varieties]]
[[Category:Theorems in geometry|Cube]]