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}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==