Appell–Humbert theorem

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) Contents 1 Statement 2 Ample line bundles 3 See also 4 References Statement 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 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 Complex torus for a treatment of the theorem with examples 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 Categories: Abelian varietiesTheorems in algebraic geometryTheorems in complex geometry