Bilinear cross-effect of a line bundle on an abelian variety ($D_L(f,g)=(f+g{)}^{\ast }L\otimes (f^{\ast }L{)}^{-1}\otimes (g^{\ast }L{)}^{-1}$)

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 $\hat{f}{\varphi }_{L}g+\hat{g}{\varphi }_{L}f$.