# Appell–Humbert theorem In matematica, 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 matematica, il 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) e Humbert (1893), and in general by Lefschetz (1921)

Índice

## Dichiarazione[]

Supporre che

$"{displaystyle$ is a complex torus given by

$"{displaystyle$displaystyle V/Lambda } dove

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

$"{displaystyle$ . Se

$"{displaystyle$ is a Hermitian form on

$"{displaystyle$ whose imaginary part

$"{displaystyle$ is integral on

$"{displaystyle$displaystyle Lambda times Lambda } , e

$"{displaystyle$ is a map from

$"{displaystyle$ to the unit circle

$"{displaystyle$zin mathbb {C} :|z|=1}} , called a semi-character, tale che

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

$"{displaystyle$pi H(z,tu)+H(tu,tu)pi /2} } è un 1-cocycle di

$"{displaystyle$ 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 ,u(1))cong mathbb {R} ^{2n}/mathbb {Z} ^{2n}} Se

$"{displaystyle$displaystyle Lambda cong mathbb {Z} ^{2n}} since any such character factors through

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

$"{displaystyle$pi ilangle l^{*},-sonaglio )} 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. Infatti, this torus can be equipped with a complex structure, giving the dual complex torus.

Esplicitamente, a line bundle on

$"{displaystyle$displaystyle T=V/Lambda } may be constructed by descent from a line bundle on

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

$"{displaystyle$displaystyle u^{*}{matematico {o}}_{V}a {matematico {o}}_{V}} , 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

$"{displaystyle$ e

$"{displaystyle$ 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 } ## Riferimenti[]

• 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", Transazioni dell'American Mathematical Society, Provvidenza, RI: Società matematica americana, 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", Transazioni dell'American Mathematical Society, Provvidenza, RI: Società matematica americana, 22 (4): 407–482, doi:10.2307/1988964, ISSN 0002-9947, JSTOR 1988964
• Mumford, Davide (2008) , Varietà abeliane, Tata Institute of Fundamental Research Studies in Matematica, vol. 5, Provvidenza, RI: Società matematica americana, ISBN 978-81-85931-86-9, SIG 0282985, OCLC 138290

Se vuoi conoscere altri articoli simili a Appell–Humbert theorem puoi visitare la categoria Varietà abeliane.

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