# Appell–Humbert theorem

In Mathematik, 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

Im Mathematik, das 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) und Humbert (1893), and in general by Lefschetz (1921)

Índice

## Aussage[]

Nehme an, dass

$"{displaystyle$

is a complex torus given by

$"{displaystyle$displaystyle V/Lambda }

wo

$"{displaystyle$

is a lattice in a complex vector space

$"{displaystyle$

. Wenn

$"{displaystyle$

is a Hermitian form on

$"{displaystyle$

whose imaginary part

$"{displaystyle$

is integral on

$"{displaystyle$displaystyle Lambda times Lambda }

, und

$"{displaystyle$

is a map from

$"{displaystyle$

to the unit circle

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

, called a semi-character, so dass

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

dann

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

ist ein 1-cocycle von

$"{displaystyle$

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}}

wenn

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

since any such character factors through

$"{displaystyle$

composed with the exponential map. Das ist, a character is a map of the form

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

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. In der Tat, this torus can be equipped with a complex structure, giving the dual complex torus.

Ausdrücklich, a line bundle on

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

may be constructed by descent from a line bundle on

$"{displaystyle$

(which is necessarily trivial) und ein descent data, namely a compatible collection of isomorphisms

$"{displaystyle$

, 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

$"{displaystyle$

und

$"{displaystyle$

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 }

## Verweise[]

• 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", Transaktionen der American Mathematical Society, Vorsehung, RI: Amerikanische Mathematische Gesellschaft, 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", Transaktionen der American Mathematical Society, Vorsehung, RI: Amerikanische Mathematische Gesellschaft, 22 (4): 407–482, doi:10.2307/1988964, ISSN 0002-9947, JSTOR 1988964
• Mumford, David (2008) [1970], Abelische Sorten, Tata Institute of Fundamental Research Studies in Mathematics, vol. 5, Vorsehung, RI: Amerikanische Mathematische Gesellschaft, ISBN 978-81-85931-86-9, HERR 0282985, OCLC 138290

Wenn Sie andere ähnliche Artikel wissen möchten Appell–Humbert theorem Sie können die Kategorie besuchen Abelische Sorten.

Geh hinauf

Wir verwenden eigene Cookies und Cookies von Drittanbietern, um die Benutzererfahrung zu verbessern Mehr Informationen