Variations of stability and hyper-Kähler structures on moduli of parabolic Higgs bundles

In this talk I will present contributions to the complex differential geometry of moduli spaces of parabolic Higgs bundles on compact Riemann surfaces, that were obtained during the course of my project. Such results possess a unifying theme, namely, the interplay between the natural families of hyper-Kähler structures that such moduli spaces possess, and the families of stability conditions that are necessary to construct them. The former are uniquely determined by the Kähler structure of a special sub-moduli space of half dimension. Of special interest is the case when the chosen Riemann surface has genus 0, for which the relation admits a more explicit form.