Special K\"ahler structures, cubic differentials and hyperbolic metrics

TL;DR

This paper explores the existence and classification of special K"ahler structures with isolated singularities on compact Riemann surfaces, establishing a correspondence with cubic differentials and hyperbolic metrics.

Contribution

It provides necessary and sufficient conditions for such structures on the Riemann sphere and characterizes their moduli space using a new correspondence.

Findings

Necessary conditions for existence are established.

Sufficient conditions are proven for the Riemann sphere.

The moduli space of structures with fixed singularities is characterized.

Abstract

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.