where the vertical arrows are isomorphisms, or equality. This diagram is typically called the Appell-Humbert theorem. Dual complex torus As mentioned before, a character on the lattice can be expressed as a function for some fixed dual vector . If we want to put a complex structure on the real torus of all characters, we need to start with a complex vector space which embeds into. It turns out that the complex vector space of complex antilinear maps, is isomorphic to the real dual vector space , which is part of the factorization for writing down characters. Furthermore, there is an associated lattice called the dual lattice of . Then, we can form the dual complex torus which has the special property that that dual of the dual complex torus is the original complex torus. Moreover, from the discussion above, we can identify the dual complex torus with the Picard group of by sending an anti-linear dual vector to giving the map which factors through the dual complex torus. There are other constructions of the dual complex torus using techniques from the theory of Abelian varietiespg 123-125. Essentially, taking a line bundle over a complex torus (or Abelian variety) , there is a closed subset of defined as the points of where their translations are invariant, i.e. Then, the dual complex torus can be constructed as presenting it as an isogeny. It can be shown that defining this way satisfied the universal properties of , hence is in fact the dual complex torus (or Abelian variety). Poincare bundle From the construction of the dual complex torus, it is suggested there should exist a line bundle over the product of the torus and its dual which can be used to present all isomorphism classes of degree 0 line bundles on . We can encode this behavior with the following two properties for any point giving the line bundle is a trivial line bundle where the first is the property discussed above, and the second acts as a normalization property. We can construct using the following hermitian form and the semi-character for . Showing this data constructs a line bundle with the desired properties follows from looking at the associated canonical factor of , and observing its behavior at various restrictions. See also Poincare bundle Complex Lie group Automorphic function Intermediate Jacobian Elliptic gamma function References Complex 2-dimensional tori - Gives tools