In mathematics, the Appell–Humbert theorem describes the line bundles on a complex torus or complex abelian variety. It was proved for 2-dimensional tori by and, and in general by

Statement

Suppose that T is a complex torus given by V/\Lambda where \Lambda is a lattice in a complex vector space V. If H is a Hermitian form on V whose imaginary part is integral on, and \alpha is a map from \Lambda to the unit circle , called a semi-character, such that then is a 1-cocycle of \Lambda defining a line bundle on T. For the trivial Hermitian form, this just reduces to a character. Note that the space of character morphisms is isomorphic with a real torus""if since any such character factors through \mathbb{R} composed with the exponential map. That is, a character is a map of the form""for some covector l^* \in V^*. The periodicity of for a linear f(x) gives the isomorphism of the character group with the real torus given above. In fact, this torus can be equipped with a complex structure, giving the dual complex torus. Explicitly, a line bundle on may be constructed by descent from a line bundle on V (which is necessarily trivial) and a descent data, namely a compatible collection of isomorphisms, one for each. Such isomorphisms may be presented as nonvanishing holomorphic functions on V, and for each u the expression above is a corresponding holomorphic function. The Appell–Humbert theorem says that every line bundle on T can be constructed like this for a unique choice of H and \alpha satisfying the conditions above.

Ample line bundles

Lefschetz proved that the line bundle L, associated to the Hermitian form H is ample if and only if H is positive definite, and in this case is very ample. A consequence is that the complex torus is algebraic if and only if there is a positive definite Hermitian form whose imaginary part is integral on