Appell–Humbert theorem

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Short description: 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

Suppose that [math]\displaystyle{ T }[/math] is a complex torus given by [math]\displaystyle{ V/\Lambda }[/math] where [math]\displaystyle{ \Lambda }[/math] is a lattice in a complex vector space [math]\displaystyle{ V }[/math]. If [math]\displaystyle{ H }[/math] is a Hermitian form on [math]\displaystyle{ V }[/math] whose imaginary part [math]\displaystyle{ E = \text{Im}(H) }[/math] is integral on [math]\displaystyle{ \Lambda\times\Lambda }[/math], and [math]\displaystyle{ \alpha }[/math] is a map from [math]\displaystyle{ \Lambda }[/math] to the unit circle [math]\displaystyle{ U(1) = \{z \in \mathbb{C} : |z| = 1 \} }[/math], called a semi-character, such that

[math]\displaystyle{ \alpha(u+v) = e^{i\pi E(u,v)}\alpha(u)\alpha(v)\ }[/math]

then

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

is a 1-cocycle of [math]\displaystyle{ \Lambda }[/math] defining a line bundle on [math]\displaystyle{ 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]\displaystyle{ \text{Hom}_{\textbf{Ab}}(\Lambda,U(1)) \cong \mathbb{R}^{2n}/\mathbb{Z}^{2n} }[/math]

if [math]\displaystyle{ \Lambda \cong \mathbb{Z}^{2n} }[/math] since any such character factors through [math]\displaystyle{ \mathbb{R} }[/math] composed with the exponential map. That is, a character is a map of the form

[math]\displaystyle{ \text{exp}(2\pi i \langle l^*, -\rangle ) }[/math]

for some covector [math]\displaystyle{ l^* \in V^* }[/math]. The periodicity of [math]\displaystyle{ \text{exp}(2\pi i f(x)) }[/math] for a linear [math]\displaystyle{ 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]\displaystyle{ T = V/\Lambda }[/math] may be constructed by descent from a line bundle on [math]\displaystyle{ V }[/math] (which is necessarily trivial) and a descent data, namely a compatible collection of isomorphisms [math]\displaystyle{ u^*\mathcal{O}_V \to \mathcal{O}_V }[/math], one for each [math]\displaystyle{ u \in U }[/math]. Such isomorphisms may be presented as nonvanishing holomorphic functions on [math]\displaystyle{ V }[/math], and for each [math]\displaystyle{ u }[/math] the expression above is a corresponding holomorphic function.

The Appell–Humbert theorem (Mumford 2008) says that every line bundle on [math]\displaystyle{ T }[/math] can be constructed like this for a unique choice of [math]\displaystyle{ H }[/math] and [math]\displaystyle{ \alpha }[/math] satisfying the conditions above.

Ample line bundles

Lefschetz proved that the line bundle [math]\displaystyle{ L }[/math], associated to the Hermitian form [math]\displaystyle{ H }[/math] is ample if and only if [math]\displaystyle{ H }[/math] is positive definite, and in this case [math]\displaystyle{ 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]\displaystyle{ \Lambda\times\Lambda }[/math]

See also

References



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