An integral of the form \(\int_{P_0}^P f(x,y)dx\), where \(f\) is a rational function and \(x\) and \(y\) satisfy a polynomial dependence \(p(x,y)=0\), is known as an Abelian integral. Fixing one endpoint \(P_0\) and allowing the other to vary on the Riemann surface \(p(x,y)=0\), we obtain a multifunction whose value depends on the (homology class of) the path along which we integrate.
We 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.