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 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 (which is necessarily trivial) and a descent data, namely a compatible collection of isomorphisms ${\displaystyle u^{*}{\mathcal {O}}_{V}\to {\mathcal {O}}_{V}}$, 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.

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