What is the Theorem of the Cube? - MathOverflow most recent 30 from http://mathoverflow.net 2013-06-19T00:55:21Z http://mathoverflow.net/feeds/question/290 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/290/what-is-the-theorem-of-the-cube What is the Theorem of the Cube? Chris Schommer-Pries 2009-10-11T16:59:30Z 2009-10-12T16:31:20Z <p>What is the "theorem of the cube" for abelian varieties? What is the statement and how should I think about it?</p> http://mathoverflow.net/questions/290/what-is-the-theorem-of-the-cube/298#298 Answer by Ilya Nikokoshev for What is the Theorem of the Cube? Ilya Nikokoshev 2009-10-11T18:24:25Z 2009-10-11T18:24:25Z <p>If you have a line bundle trivial on 3 "surfaces" of a "cube" <code>A x B x C</code> where A, B, C are abelian varieties, then this line bundle in trivial on the whole "cube".</p> <p>See <a href="http://en.wikipedia.org/wiki/Theorem%20of%20the%20cube" rel="nofollow">wikipedia</a>.</p> http://mathoverflow.net/questions/290/what-is-the-theorem-of-the-cube/356#356 Answer by David Zureick-Brown for What is the Theorem of the Cube? David Zureick-Brown 2009-10-12T16:31:20Z 2009-10-12T16:31:20Z <p>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 <a href="http://math.berkeley.edu/~molsson/Hangzhou-workshop-notes.pdf" rel="nofollow">notes</a> from this summer's Hangzhou workshop.</p>