Theorem of the Cube

The Theorem of the Cube says that the third finite difference of a line bundle on an abelian variety is trivial. On $X^3$, it is the alternating tensor product of the pullbacks along the seven nonempty partial-sum maps.

For a line bundle on an elliptic curve over a number field, the geometric cube identity remains exact as a line-bundle identity. The Weil height machine translates it into the displayed bounded-error identity for a chosen logarithmic height, uniformly on the rational triples. This distinction is essential: arbitrary naive heights do not obey an exact quadratic identity. Dividing the identity for $2^{n}P,2^{n}Q,2^{n}R$ by $4^n$ produces an exact identity for the canonical height of an elliptic curve.

If a line bundle on $E^3$ is trivial on each zero-coordinate face, its First Chern class vanishes: the integral second cohomology consists of single-factor second cohomology and cross-products of first cohomology from pairs, and each summand is detected on a coordinate face. The holomorphic exponential sequence then puts the bundle in ${\mathrm{Pic}}^0(E^3)$. The decomposition of first cohomology, including Dolbeault cohomology, identifies this group with ${\mathrm{Pic}}^0(E{)}^3$. Triviality on the three coordinate axes makes all three factors trivial. Applying this cube principle to the alternating partial-sum pullbacks proves the usual addition-map Theorem of the Cube.

For group homomorphisms $f,g:A\to B$, this Picard group class is symmetric and additive in both arguments. The Theorem of the Cube says precisely that the third additive difference of $h\mapsto [h^{\ast }L]$ vanishes, proving additivity of the cross-effect. The Poincaré line bundle expresses it as $(f,{\varphi }_{L}g{)}^{\ast }{P}_B$, so it depends only on the Néron-Severi group class of $L$. Its associated homomorphism is $\hat{f}{\varphi }_{L}g+\hat{g}{\varphi }_{L}f$.

For a line bundle $L$ on an abelian variety $A$ and translations $T_x(a)=a+x$, the Theorem of the square gives an isomorphism of line bundlesConsequently $x\mapsto T_x^{\ast }L\otimes L^{-1}$ is a group homomorphism from $A$ to its dual abelian variety. These are isomorphism classes: no canonical trivialization is supplied without normalization data. On an elliptic curve, translating a degree-one divisor changes its degree-zero difference class additively, illustrating the identity. The Theorem of the Cube controls the additional compatibility when the translation points themselves vary.^[ref]

For a line bundle $L$ on an abelian variety $A$, setThe translated difference is algebraically trivial, so its value belongs to the dual abelian variety $\hat{A}={\mathrm{Pic}}^0(A)$ rather than merely the whole Picard group. The Theorem of the square makes this a group homomorphism. Tensoring line bundles adds their homomorphisms; an algebraically trivial bundle gives the zero map. An ample line bundle gives a finite surjective map, a polarization of an abelian variety; its kernel measures translations preserving the bundle class.

For a line bundle $L$ on an abelian variety $A$, this closed subgroup consists of $a$ such that $T_a^{\ast }L\cong L$. It is a stabilizer of a Picard group class, which need not be the same as the stabilizer of an individual Weil divisor representing that class.

For an abelian variety $A$, the Identity component of the Picard group ${\mathrm{Pic}}^0(A)$ consists of algebraically trivial line bundles and is represented by the dual abelian variety. In terms of translations, it isThus these bundles are translation-invariant as isomorphism classes. This does not say their individual global sections are invariant, nor that every such bundle is trivial. On an elliptic curve, $\mathcal{O}(p-q)$ is a degree-zero bundle; it is trivial exactly when its class $p-q$ is the identity.

A useful form of the Seesaw theorem determines a line bundle on a product from its fibre restrictions. Let $X$ be a complete geometrically integral variety with a base point $x_0$, and let $L$ be a line bundle on $X\times Y$ trivial on each geometric fibre $X\times \{y\}$. For a reduced variety $Y$, constancy of the fibrewise section dimension and cohomology and base change imply that $L$ is pulled back from $Y$; restricting to $\{x_0\}\times Y$ identifies the bundle being pulled back. If that transverse restriction is also trivial, then $L$ is trivial. This is useful for verifying normalized universal bundle identities by checking fibres; matching only one fibre in each direction is insufficient.