Abelian integrals and categoricity

An integral of the form $\int w(z)dz$ where $w(z)$ is an algebraic function of $z$, such as $\int\frac{dz}{\sqrt{z^{3}-1}}$, is known as an Abelian integral. Once represented as a path integral of a rational differential form on a complex algebraic curve, it can be seen as a multifunction of the endpoints whose value depends on (the homology class of) the path along which the integral is taken.

I will consider the model-theoretic status of such multifunctions, and in particular the problem of giving categorical elementary descriptions of structures incorporating them and their interactions with the complex field. Following work by Zilber and Gavrilovich, we will find that the classical theory of Abelian varieties, along with Faltings' work and some model theoretic ideas due to Shelah, allow us to give partially satisfactory answers in some special cases.