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 Appell (1891) and Humbert (1893), and in general by Lefschetz (1921) Suppose that

is a complex torus given by

is a lattice in a complex vector space

is a Hermitian form on

whose imaginary part

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

since any such character factors through

composed with the exponential map.

That is, a character is a map of the form

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

(which is necessarily trivial) and a descent data, namely a compatible collection of isomorphisms

Such isomorphisms may be presented as nonvanishing holomorphic functions on

the expression above is a corresponding holomorphic function.

The Appell–Humbert theorem (Mumford 2008) says that every line bundle on

can be constructed like this for a unique choice of

Lefschetz proved that the line bundle

, associated to the Hermitian form

is positive definite, and in this case

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