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 is a complex torus given by where is a lattice in a complex vector space. If is a Hermitian form on whose imaginary part is integral on, and is a map from to the unit circle, called a semi-character, such that
then
is a 1-cocycle of defining a line bundle on. 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 composed with the exponential map. That is, a character is a map of the form

for some covector. The periodicity of for a linear 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 and a descent data, namely a compatible collection of isomorphisms, one for each. Such isomorphisms may be presented as nonvanishing holomorphic functions on, and for each the expression above is a corresponding holomorphic function.
The Appell–Humbert theorem says that every line bundle on can be constructed like this for a unique choice of and satisfying the conditions above.

Ample line bundles

Lefschetz proved that the line bundle, associated to the Hermitian form is ample if and only if 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