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)
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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 on U defining a line bundle on T.
The Appell–Humbert theorem (Mumford 2008) 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.