Theorem of the cube

In algebraic geometry, the theorem of the cube is a condition for a line bundle over a product of three complete varieties to be trivial. It was first discovered in the context of linear equivalence by the Italian school of algebraic geometry. The modern version of the theorem was first published by Serge Lang, who credited it to André Weil. A treatment by means of sheaf cohomology, and description in terms of the Picard functor, was given by David Mumford.

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 $U\times V\times \{w\}$, $U\times \{v\}\times W$, and $\{u\}\times V\times W$ is itself trivial.

Special cases

On a ringed space $X$, an invertible sheaf $L$ is trivial if it is isomorphic to ${{𝒪}}_X$ as an ${{𝒪}}_X$-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.

Theorem of the square

The theorem of the square is a corollary (also due to Weil) applying to an abelian variety $A$. One version of it states that the function ${\varphi }_L$ taking $x\in A$ to $T_x^{*}L\otimes L^{-1}$ is a group homomorphism from $A$ to $\mathrm{Pic}(A)$ (where $T_x^{*}$ is translation by $x$ on line bundles).

References

• Kleiman, Steven L. (2005). "The Picard scheme". Fundamental Algebraic Geometry. Mathematical Surveys and Monographs. 123. Providence: American Mathematical Society. pp. 235–321. ISBN 978-0-8218-4245-4. 

• Lang, Serge (1983). Abelian Varieties. New York: Springer. ISBN 978-0-387-90875-5. 

• Mumford, David (1970). Abelian Varieties. Tata Institute of Fundamental Research Studies in Mathematics. 5. Oxford University Press. 

• Polishchuk, Alexander (2003). Abelian Varieties, Theta Functions and the Fourier Transform. Cambridge Tracts in Mathematics. 153. Cambridge University Press. ISBN 978-0-52180804-0.