Theorem of the cube: Difference between revisions
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by {{harvtxt|Mumford|2008}}. |
by {{harvtxt|Mumford|2008}}. |
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==Statement== |
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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'' |
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.) |
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===Special cases=== |
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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. |
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| ⚫ | 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 |
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===Restatement using biextensions=== |
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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> |
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> |
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==Theorem of the square== |
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| ⚫ | 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). |
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==References== |
==References== |
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[[Category:Abelian varieties]] |
[[Category:Abelian varieties]] |
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[[Category:Algebraic varieties]] |
[[Category:Algebraic varieties]] |
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[[Category:Theorems in geometry]] |
[[Category:Theorems in geometry|Cube]] |
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Latest revision as of 03:06, 29 September 2024
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
[edit]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
[edit]On a ringed space X, an invertible sheaf L is trivial if isomorphic to OX, as an OX-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
[edit]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
[edit]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 φL 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
[edit]- 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
[edit]- ^ Alexander Polishchuk, Abelian Varieties, Theta Functions and the Fourier Transform (2003), p. 122.