MathOverflow is a question and answer site for professional mathematicians. It's 100% free, no registration required.

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

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

share|cite|improve this question
up vote 7 down vote accepted

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

See wikipedia.

share|cite|improve this answer
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.

share|cite|improve this answer
+1, it's very informative, thanks! – Ilya Nikokoshev Oct 12 '09 at 17:37

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.