We obtain necessary conditions for the existence of special K\"ahler structures with isolated singularities on compact Riemann surfaces. We prove that these conditions are also sufficient in the case of the Riemann sphere and, moreover, we determine the whole moduli space of special K\"ahler structures with fixed singularities. The tool we develop for this aim is a correspondence between special K\"ahler structures and pairs consisting of a cubic differential and a hyperbolic metric.
Categories: math.DG math.AG