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)

Aussage[]

Nehme an, dass

is a complex torus given by

displaystyle V/Lambda }

wo

is a lattice in a complex vector space

. Wenn

is a Hermitian form on

whose imaginary part

is integral on

displaystyle Lambda times Lambda }

, und

is a map from

to the unit circle

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

, called a semi-character, so dass

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

dann

pi H(z,u)+H(u,u)Pi /2} }

ist ein 1-cocycle von

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}}

wenn

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

since any such character factors through

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

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

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

Ausdrücklich, a line bundle on

displaystyle T=V/Lambda }

may be constructed by descent from a line bundle on

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

, one for each

displaystyle uin U}

. 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

und

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 }

Siehe auch[]

• Complex torus for a treatment of the theorem with examples

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