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 and, and in general by

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


 * $$\alpha(u+v) = e^{i\pi E(u,v)}\alpha(u)\alpha(v)\ $$

then


 * $$ \alpha(u)e^{\pi H(z,u)+H(u,u)\pi/2}\ $$

is a 1-cocycle of $$\Lambda$$ defining a line bundle on $$T$$. For the trivial Hermitian form, this just reduces to a character. Note that the space of character morphisms is isomorphic with a real torus"$\text{Hom}_{\textbf{Ab}}(\Lambda,U(1)) \cong \mathbb{R}^{2n}/\mathbb{Z}^{2n}$|undefined"if $$\Lambda \cong \mathbb{Z}^{2n}$$ since any such character factors through $$\mathbb{R}$$ composed with the exponential map. That is, a character is a map of the form"$\text{exp}(2\pi i \langle l^*, -\rangle )$"for some covector $$l^* \in V^*$$. The periodicity of $$\text{exp}(2\pi i f(x))$$ for a linear $$f(x)$$ 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 $$T = V/\Lambda$$ may be constructed by descent from a line bundle on $$V$$ (which is necessarily trivial) and a descent data, namely a compatible collection of isomorphisms $$u^*\mathcal{O}_V \to \mathcal{O}_V$$, one for each $$u \in \Lambda$$. Such isomorphisms may be presented as nonvanishing holomorphic functions on $$V$$, and for each $$u$$ the expression above is a corresponding holomorphic function.

The Appell–Humbert theorem says that every line bundle on $$T$$ can be constructed like this for a unique choice of $$H$$ and $$\alpha$$ satisfying the conditions above.

Ample line bundles
Lefschetz proved that the line bundle $$L$$, associated to the Hermitian form $$H$$ is ample if and only if $$H$$ is positive definite, and in this case $$L^{\otimes 3}$$ 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 $$\Lambda\times\Lambda$$