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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 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 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, 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 φ_L 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., vol. 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, vol. 7, New York: Interscience Publishers, Inc., MR 0106225
Mumford, David (2008) [1970], Abelian varieties, Tata Institute of Fundamental Research Studies in Mathematics, vol. 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.

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

[1]

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-In [[mathematics]], the '''theorem of the cube''' is a foundational result in the [[algebraic geometry]] of a [[complete variety]]. It was a principle discovered, in the context of [[linear equivalence]], by the [[Italian school of algebraic geometry]]. The specific result was proved under this name, in the early 1950s, in the course of his fundamental work on abstract algebraic geometry by [[André Weil]]; a discussion of the history has been given by [[Steven Kleiman]] (lectures ''The Picard Scheme'', Introduction) . A treatment by means of [[sheaf cohomology]], and description in terms of the [[Picard functor]], was given in ''Abelian Varieties'' (1970) by [[David Mumford]].
+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 varieties ''U'', ''V'' and ''W'', and given points ''u'', ''v'' and ''w'' on them, any [[invertible sheaf]] ''L'' which has a trivial restriction to each of ''U''&times; ''V'' &times; {''w''}, ''U''&times; {''v''} &times; ''W'', and {''u''} &times; ''V'' &times; ''W'', is itself trivial. (Mumford p. 55; the result there is slightly stronger, in that one of the varieties need not be complete.)
+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===
-Note: On a [[ringed space]] ''X'', an invertible sheaf ''L'' is ''trivial'' if isomorphic to ''O''<sub>''X''</sub>, as ''O''<sub>''X''</sub>- module. If ''L'' is taken as a [[holomorphic line bundle]], in the [[complex manifold]] case, this is the same here as a [[trivial bundle]], but in a ''holomorphic'' sense, not just topologically.
+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===
-The '''theorem of the square''' (Mumford p.59) is a corollary applying to an [[abelian variety]] ''A'', defining a group homomorphism from ''A'' to ''Pic''(''A''), in terms of the change in ''L'' by translation on ''A''.
+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:Mathematical theorems]]
+[[Category:Theorems in geometry|Cube]]