# 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

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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

$$is a complex torus given by

$$where

$$is a lattice in a complex vector space

$$. If

$$is a Hermitian form on

$$whose imaginary part

${\scriptscriptstyle E=\text{Im}\left(H\right)}$is integral on

$$, and

$$is a map from

$$to the unit circle

$$, called a **semi-character**, such that

- $$

then

- $$

is a 1-cocycle of

$$defining a line bundle on

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

$$

if

$$since any such character factors through

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

$$

for some covector

$$. The periodicity of

$$for a linear

$$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

$$may be constructed by descent from a line bundle on

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

$$, one for each

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

$$, and for each

$$the expression above is a corresponding holomorphic function.

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

$$can be constructed like this for a unique choice of

$$and

$$satisfying the conditions above.

## Ample line bundles[]

Lefschetz proved that the line bundle

$$, associated to the Hermitian form

$$is ample if and only if

$$is positive definite, and in this case

$$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

$$## See also[]

- Complex torus for a treatment of the theorem with examples

## 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, 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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