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)

Statement

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Suppose that   is a complex torus given by   where   is a lattice in a complex vector space  . If   is a Hermitian form on   whose imaginary part   is integral on  , and   is a map from   to the unit circle  , called a semi-character, such that

 

then

 

is a 1-cocycle of   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

 

if   since any such character factors through   composed with the exponential map. That is, a character is a map of the form

 

for some covector  . The periodicity of   for a linear   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   may be constructed by descent from a line bundle on   (which is necessarily trivial) and a descent data, namely a compatible collection of isomorphisms  , 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   and   satisfying the conditions above.

Ample line bundles

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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  

See also

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References

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  • 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, 22 (3), Providence, R.I.: American Mathematical Society: 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, 22 (4), Providence, R.I.: American Mathematical Society: 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