Analytic invariants for stability variations on moduli of parabolic bundles

A characteristic of moduli spaces of parabolic bundles on Riemann surfaces is their dependence on a set of real parameters, which gives rise to wall-crossing phenomena in the study of their birational geometry. In a less explicit way, this dependance also occurs in their natural Kähler metrics, and it is compelling to express this dependence as a suitable "Torelli theorem" answering the question: to what extent does the Kähler metric (an analytic invariant) determine the stability conditions used to define the moduli problem?

In this talk I will describe a strategy to address this variation problem in terms of complex-analytic techniques arising in the study of the spectral geometry of resolvents. I will also explain how these techniques can be applied to the study and classification of gravitational instantons of ALG type. This is work in progress, joint with Hartmut Weiss.