Английская Википедия:Appell–Humbert theorem
Шаблон:Short description 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 Шаблон:Harvs and Шаблон:Harvs, and in general by Шаблон:Harvs
Statement
Suppose that <math>T</math> is a complex torus given by <math>V/\Lambda</math> where <math>\Lambda</math> is a lattice in a complex vector space <math>V</math>. If <math>H</math> is a Hermitian form on <math>V</math> whose imaginary part <math>E = \text{Im}(H)</math> is integral on <math>\Lambda\times\Lambda</math>, and <math>\alpha</math> is a map from <math>\Lambda</math> to the unit circle <math>U(1) = \{z \in \mathbb{C} : |z| = 1 \}</math>, called a semi-character, such that
- <math>\alpha(u+v) = e^{i\pi E(u,v)}\alpha(u)\alpha(v)\ </math>
then
- <math> \alpha(u)e^{\pi H(z,u)+H(u,u)\pi/2}\ </math>
is a 1-cocycle of <math>\Lambda</math> defining a line bundle on <math>T</math>. For the trivial Hermitian form, this just reduces to a character. Note that the space of character morphisms is isomorphic with a real torus
<math>\text{Hom}_{\textbf{Ab}}(\Lambda,U(1)) \cong \mathbb{R}^{2n}/\mathbb{Z}^{2n}</math>
if <math>\Lambda \cong \mathbb{Z}^{2n}</math> since any such character factors through <math>\mathbb{R}</math> composed with the exponential map. That is, a character is a map of the form
<math>\text{exp}(2\pi i \langle l^*, -\rangle )</math>
for some covector <math>l^* \in V^*</math>. The periodicity of <math>\text{exp}(2\pi i f(x))</math> for a linear <math>f(x)</math> 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 <math>T = V/\Lambda</math> may be constructed by descent from a line bundle on <math>V</math> (which is necessarily trivial) and a descent data, namely a compatible collection of isomorphisms <math>u^*\mathcal{O}_V \to \mathcal{O}_V</math>, one for each <math>u \in U</math>. Such isomorphisms may be presented as nonvanishing holomorphic functions on <math>V</math>, and for each <math>u</math> the expression above is a corresponding holomorphic function.
The Appell–Humbert theorem Шаблон:Harv says that every line bundle on <math>T</math> can be constructed like this for a unique choice of <math>H</math> and <math>\alpha</math> satisfying the conditions above.
Ample line bundles
Lefschetz proved that the line bundle <math>L</math>, associated to the Hermitian form <math>H</math> is ample if and only if <math>H</math> is positive definite, and in this case <math>L^{\otimes 3}</math> 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 <math>\Lambda\times\Lambda</math>
See also
- Complex torus for a treatment of the theorem with examples
References
