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)

Dichiarazione[]

Supporre che

is a complex torus given by

displaystyle V/Lambda }

dove

is a lattice in a complex vector space

. Se

is a Hermitian form on

whose imaginary part

is integral on

displaystyle Lambda times Lambda }

, e

is a map from

to the unit circle

zin mathbb {C} :|z|=1}}

, called a semi-character, tale che

u+v)=e^{ipi E(tu,v)}alfa (tu)alfa (v) }

poi

pi H(z,tu)+H(tu,tu)pi /2} }

è un 1-cocycle di

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

Hom}}_{textbf {Ab}}(Lambda ,u(1))cong mathbb {R} ^{2n}/mathbb {Z} ^{2n}}

Se

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

since any such character factors through

composed with the exponential map. Questo è, a character is a map of the form

pi ilangle l^{*},-sonaglio )}

for some covector

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

. The periodicity of

pi if(X))}

for a linear

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 T=V/Lambda }

may be constructed by descent from a line bundle on

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

displaystyle u^{*}{matematico {o}}_{V}a {matematico {o}}_{V}}

, 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

e

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

displaystyle Lambda times Lambda }

Guarda anche[]

• Complex torus for a treatment of the theorem with examples

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) [1970], 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