 Appell–Humbert theorem

For a complex torus $X=V/\Lambda$, every holomorphic line bundle is classified by a pair $(H,\alpha )$. Here $H$ is a Hermitian form, linear in its first argument, whose imaginary part $E$ is integral on $\Lambda \times \Lambda$, and $\alpha$ is a semicharacter of a complex lattice:The corresponding bundle $L(H,\alpha )$ is the quotient of $V\times \mathbb{C}$ by the group actionThe semicharacter identity makes successive lattice translations compose correctly. The resulting correspondence is an isomorphism onto the Picard group, so it gives both existence and uniqueness of the classification.^[ref]

Tensor products add the Hermitian forms and multiply the semicharacters:When $H=0$, the semicharacter is an ordinary unitary group homomorphism, describing bundles with zero First Chern class. A positive-definite $H$ instead gives a polarization of a complex torus. The theorem therefore separates the discrete integral curvature data from the remaining flat holomorphic choices.

The Appell–Humbert theorem is a result in the theory of complex numbers and multidimensional analysis. It relates to the behavior of certain classes of functions, particularly those that are harmonic or analytic. The theorem states conditions for when a function can be expressed as a series of its values on a certain domain.