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What is the "theorem of the cube" for abelian varieties? What is the statement and how should I think about it?

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up vote 6 down vote accepted

If you have a line bundle trivial on 3 "surfaces" of a "cube" A x B x C where A, B, C are abelian varieties, then this line bundle in trivial on the whole "cube".

See wikipedia.

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NB: it's valid not only for abelian varieties, but for any complete varieties... –  Michael Thaddeus Jun 7 '10 at 11:52

One application of the theorem of the cube is to study the map from an abelian variety A to its dual abelian variety; the map is defined in terms of line bundles and the key technical theorem one uses to prove anything (e.g. that the map to the dual is a homomorphism) is the theorem of the cube. See Mumford's Abelian Varieties book or Martin Olsson's notes from this summer's Hangzhou workshop.

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+1, it's very informative, thanks! –  Ilya Nikokoshev Oct 12 '09 at 17:37

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