Appell–Humbert theorem – Wikipedia

Describes the line bundles on a complex torus or complex abelian variety

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)

Statement[edit]

Suppose that

${displaystyle T}$

is a complex torus given by

${displaystyle V/Lambda }$

where

${displaystyle Lambda }$

is a lattice in a complex vector space

${displaystyle V}$

. If

${displaystyle H}$

is a Hermitian form on

${displaystyle V}$

whose imaginary part

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

is integral on

${displaystyle Lambda times Lambda }$

, and

${displaystyle alpha }$

is a map from

${displaystyle Lambda }$

to the unit circle

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

, 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

${displaystyle Lambda }$

defining a line bundle on

${displaystyle T}$

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

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

if

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

since any such character factors through

${displaystyle mathbb {R} }$

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

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

for some covector

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

. The periodicity of

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

for a linear

${displaystyle f(x)}$

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

${displaystyle T=V/Lambda }$

may be constructed by descent from a line bundle on

${displaystyle V}$

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

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

, one for each

${displaystyle uin U}$

. Such isomorphisms may be presented as nonvanishing holomorphic functions on

${displaystyle V}$

, and for each

${displaystyle u}$

the expression above is a corresponding holomorphic function.

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

${displaystyle T}$

can be constructed like this for a unique choice of

${displaystyle H}$

and

${displaystyle alpha }$

satisfying the conditions above.

Ample line bundles[edit]

Lefschetz proved that the line bundle

${displaystyle L}$

, associated to the Hermitian form

${displaystyle H}$

is ample if and only if

${displaystyle H}$

is positive definite, and in this case

${displaystyle L^{otimes 3}}$

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

${displaystyle Lambda times Lambda }$

See also[edit]

References[edit]

• 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