theorem of the cube
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 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{{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=
Weil's result has been restated in terms of biextensions, a concept now generally used in the duality theory of abelian varieties.Alexander Polishchuk, Abelian Varieties, Theta Functions and the Fourier Transform (2003), p. 122.
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 | orig-date=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}}