Appell–Humbert theorem


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/U where U is a lattice in a complex vector space V. If H is a Hermitian form on V whose imaginary part E is integral on U×U, and α is a map from U to the unit circle such that
then
is a 1-cocycle of U defining a line bundle on T. Explicitly, a line bundle on T = V/U may be constructed by descent from a line bundle on V and a descent data, namely a compatible collection of isomorphisms, one for each u ∈ U. 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 α 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 L3 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 U×U.