# Théorème d'Appell-Humbert

En mathématiques, 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)

# Théorème d'Appell-Humbert

Dans mathématiques, la Théorème d'Appell-Humbert describes the line bundles on a complex torus or complex abelian variety.
It was proved for 2-dimensional tori by Appell (1891) et Humbert (1893), and in general by Lefschetz (1921)

Índice

## Déclaration[]

Supposer que

$"{displaystyle$

is a complex torus given by

$"{displaystyle$displaystyle V/Lambda }

$"{displaystyle$

is a lattice in a complex vector space

$"{displaystyle$

. Si

$"{displaystyle$

is a Hermitian form on

$"{displaystyle$

whose imaginary part

$"{displaystyle$

is integral on

$"{displaystyle$displaystyle Lambda times Lambda }

, et

$"{displaystyle$

is a map from

$"{displaystyle$

to the unit circle

$"{displaystyle$zin mathbb {C} :|z|=1}}

, called a semi-character, tel que

$"{displaystyle$u+v)=e^{ipi E(tu,v)}alpha (tu)alpha (v) }

alors

$"{displaystyle$pi H(z,tu)+H(tu,tu)pi /2} }

est un 1-cocycle de

$"{displaystyle$

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

$"{displaystyle$Hom}}_{textbf {Ab}}(Lambda ,tu(1))cong mathbb {R} ^{2n}/mathbb {Z} ^{2n}}

si

$"{displaystyle$displaystyle Lambda cong mathbb {Z} ^{2n}}

since any such character factors through

$"{displaystyle$

composed with the exponential map. C'est-à-dire, a character is a map of the form

$"{displaystyle$pi ilangle l^{*},-hochet )}

for some covector

$"{displaystyle$displaystyle l^{*}in V^{*}}

. The periodicity of

$"{displaystyle$pi if(X))}

for a linear

$"{displaystyle$

gives the isomorphism of the character group with the real torus given above. En réalité, this torus can be equipped with a complex structure, giving the dual complex torus.

Explicitement, a line bundle on

$"{displaystyle$displaystyle T=V/Lambda }

may be constructed by descent from a line bundle on

$"{displaystyle$

(which is necessarily trivial) et un descent data, namely a compatible collection of isomorphisms

$"{displaystyle$displaystyle u^{*}{mathématique {O}}_{V}à {mathématique {O}}_{V}}

, one for each

$"{displaystyle$displaystyle uin U}

. 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

$"{displaystyle$

et

$"{displaystyle$

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

$"{displaystyle$displaystyle Lambda times Lambda }

## Références[]

• 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 de l'American Mathematical Society, Providence, R.I.: Société mathématique américaine, 22 (3): 327–406, est ce que je:10.2307/1988897, ISSN 0002-9947, JSTOR 1988897
• Lefschetz, Solomon (1921), "On Certain Numerical Invariants of Algebraic Varieties with Application to Abelian Varieties", Transactions de l'American Mathematical Society, Providence, R.I.: Société mathématique américaine, 22 (4): 407–482, est ce que je:10.2307/1988964, ISSN 0002-9947, JSTOR 1988964
• Mumford, David (2008) [1970], Variétés abéliennes, Institut Tata d'études de recherche fondamentale en mathématiques, vol. 5, Providence, R.I.: Société mathématique américaine, ISBN 978-81-85931-86-9, M 0282985, OCLC 138290

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