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

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

then

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

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.

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

• Appell, P. (1891), "Sur les functiones périodiques de deux variables", Journal de Mathématiques Pures et Appliquées, Série IV, 7: 157–219
• Humbert, G. (1893), "Théorie générale des surfaces hyperelliptiques", Journal de Mathématiques Pures et Appliquées, Série IV, 9: 29–170, 361–475
• Lefschetz, Solomon (1921), "On Certain Numerical Invariants of Algebraic Varieties with Application to Abelian Varieties", Transactions of the American Mathematical Society, Providence, R.I.: American Mathematical Society, 22 (3): 327–406, doi:10.2307/1988897, ISSN 0002-9947, JSTOR 1988897
• Lefschetz, Solomon (1921), "On Certain Numerical Invariants of Algebraic Varieties with Application to Abelian Varieties", Transactions of the American Mathematical Society, Providence, R.I.: American Mathematical Society, 22 (4): 407–482, doi:10.2307/1988964, ISSN 0002-9947, JSTOR 1988964
• Mumford, David (2008) [1970], Abelian varieties, Tata Institute of Fundamental Research Studies in Mathematics, 5, Providence, R.I.: American Mathematical Society, ISBN 978-81-85931-86-9, MR 0282985, OCLC 138290