Gerbes and the holomorphic Brauer group of complex tori

The purpose of this paper is to develop the theory of holomorphic gerbes on complex tori in a manner analogous to the classical theory for line bundles. In contrast to past studies on this subject, we do not restrict to the case where these gerbes are torsion or topologically trivial. We give an Appell-Humbert type description of all holomorphic gerbes on complex tori. This gives an explicit, simple, cocycle representative (and hence gerbe) for each equivalence class of holomorphic gerbes. We also prove that a gerbe on the fiber product of four spaces over a common base is trivial as long as it is trivial upon restriction to any three out of the four spaces. A fine moduli stack for gerbes on complex tori is constructed. This involves the construction of a "Poincaré" gerbe which plays a role analogous to the role of the Poincaré bundle in the case of line bundles.