Appell–Humbert theorem

In mathematics, 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

Jump to navigation
Jump to search

In mathematics, 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)

Índice

Contents
Statement[]
Ample line bundles[]
See also[]
References[]

Statement[]

Suppose that

{displaystyle T}

$"T"$ is a complex torus given by

{displaystyle V/Lambda }

$"{displaystyle$ where

{displaystyle Lambda }

$"Lambda$ is a lattice in a complex vector space

{displaystyle V}

$"V"$ . If

{displaystyle H}

$"H"$ is a Hermitian form on

{displaystyle V}

$"V"$ whose imaginary part

{displaystyle E={text{Im}}(H)}

$"{displaystyle$ is integral on

{displaystyle Lambda times Lambda }

$"{displaystyle$ , and

{displaystyle alpha }

$"alpha$ is a map from

{displaystyle Lambda }

$"Lambda$ to the unit circle

{displaystyle U(1)={zin mathbb {C} :|z|=1}}

$"{displaystyle$ , called a semi-character, such that

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

then

{displaystyle alpha (u)e^{pi H(z,u)+H(u,u)pi /2} }

is a 1-cocycle of

{displaystyle Lambda }

$"Lambda$ defining a line bundle on

{displaystyle T}

$"T"$ . For the trivial Hermitian form, this just reduces to a character. Note that the space of character morphisms is isomorphic with a real torus

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

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

$"{displaystyle$ since any such character factors through

{displaystyle mathbb {R} }

$"mathbb$ composed with the exponential map. That is, a character is a map of the form

${displaystyle {text{exp}}(2pi ilangle l^{*},-rangle )}$
$"{displaystyle$

for some covector

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

$"{displaystyle$ . The periodicity of

{displaystyle {text{exp}}(2pi if(x))}

$"{displaystyle$ for a linear

{displaystyle f(x)}

$"f(x)"$ gives the isomorphism of the character group with the real torus given above. In fact, this torus can be equipped with a complex structure, giving the dual complex torus.

Explicitly, a line bundle on

{displaystyle T=V/Lambda }

$"{displaystyle$ may be constructed by descent from a line bundle on

{displaystyle V}

$"V"$ (which is necessarily trivial) and a descent data, namely a compatible collection of isomorphisms

{displaystyle u^{*}{mathcal {O}}_{V}to {mathcal {O}}_{V}}

$"{displaystyle$ , one for each

{displaystyle uin U}

$"u$ . Such isomorphisms may be presented as nonvanishing holomorphic functions on

{displaystyle V}

$"V"$ , and for each

{displaystyle u}

$"u"$ the expression above is a corresponding holomorphic function.

The Appell–Humbert theorem (Mumford 2008) says that every line bundle on

{displaystyle T}

$"T"$ can be constructed like this for a unique choice of

{displaystyle H}

$"H"$ and

{displaystyle alpha }

$"alpha$ satisfying the conditions above.

Ample line bundles[]

Lefschetz proved that the line bundle

{displaystyle L}

$"L"$ , associated to the Hermitian form

{displaystyle H}

$"H"$ is ample if and only if

{displaystyle H}

$"H"$ is positive definite, and in this case

{displaystyle L^{otimes 3}}

$"{displaystyle$ 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 Lambda times Lambda }

$"{displaystyle$

References[]

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 of the American Mathematical Society, Providence, R.I.: American Mathematical Society, 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", Transactions of the American Mathematical Society, Providence, R.I.: American Mathematical Society, 22 (4): 407–482, doi:10.2307/1988964, ISSN 0002-9947, JSTOR 1988964
Mumford, David (2008) [1970], Abelian varieties, Tata Institute of Fundamental Research Studies in Mathematics, vol. 5, Providence, R.I.: American Mathematical Society, ISBN 978-81-85931-86-9, MR 0282985, OCLC 138290

Retrieved from "https://en.wikipedia.org/w/index.php?title=Appell–Humbert_theorem&oldid=1034959685"

Categories:

Hidden categories:

Si quieres conocer otros artículos parecidos a Appell–Humbert theorem puedes visitar la categoría Abelian varieties.

Deja una respuesta Cancelar la respuesta

Appell–Humbert theorem

Appell–Humbert theorem

Contents

Statement[]

Ample line bundles[]

See also[]

References[]