# 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)

# 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)

Índice

## Statement[]

Suppose that

$"{displaystyle$ is a complex torus given by

$"{displaystyle$ where

$"{displaystyle$ is a lattice in a complex vector space

$"{displaystyle$ . If

$"{displaystyle$ is a Hermitian form on

$"{displaystyle$ whose imaginary part

$"{displaystyle$ is integral on

$"{displaystyle$ , and

$"{displaystyle$ is a map from

$"{displaystyle$ to the unit circle

$"{displaystyle$ , called a semi-character, such that

$"{displaystyle$ then

$"{displaystyle$ is a 1-cocycle of

$"{displaystyle$ defining a line bundle on

$"{displaystyle$ . For the trivial Hermitian form, this just reduces to a character. Note that the space of character morphisms is isomorphic with a real torus

$"{displaystyle$ if

$"{displaystyle$ since any such character factors through

$"{displaystyle$ composed with the exponential map. That is, a character is a map of the form

$"{displaystyle$ for some covector

$"{displaystyle$ . The periodicity of

$"{displaystyle$ for a linear

$"{displaystyle$ 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

$"{displaystyle$ may be constructed by descent from a line bundle on

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

$"{displaystyle$ , one for each

$"{displaystyle$ . Such isomorphisms may be presented as nonvanishing holomorphic functions on

$"{displaystyle$ , and for each

$"{displaystyle$ the expression above is a corresponding holomorphic function.

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

$"{displaystyle$ can be constructed like this for a unique choice of

$"{displaystyle$ and

$"{displaystyle$ satisfying the conditions above.

## Ample line bundles[]

Lefschetz proved that the line bundle

$"{displaystyle$ , associated to the Hermitian form

$"{displaystyle$ is ample if and only if

$"{displaystyle$ is positive definite, and in this case

$"{displaystyle$ 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

$"{displaystyle$ ## 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) , Abelian varieties, Tata Institute of Fundamental Research Studies in Mathematics, vol. 5, Providence, R.I.: American Mathematical Society, ISBN 978-81-85931-86-9, MR 0282985, OCLC 138290

Si quieres conocer otros artículos parecidos a Appell–Humbert theorem puedes visitar la categoría Abelian varieties.

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