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)

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

\alpha(u+v) = e^{i\pi E(u,v)}\alpha(u)\alpha(v)\

then

\alpha(u)e^{\pi H(z,u)+H(u,u)\pi/2}\

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.

References

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