Stable maps in higher dimensions

One of the basic questions in complex geometry is to understand the existence of certain canonical Kähler metrics on smooth complex manifolds or projective varieties. On Riemann surfaces one can ask for metrics of constant curvature, of prescribed curvature more generally, or of constant curvature but with prescribed singularities at points. In higher dimensions the correct analogue of constant curvature metrics are constant scalar curvature Kähler metrics, an important special case being Kähler-Einstein metrics. I will discuss the Yau-Tian-Donaldson conjecture, which relates the existence of these metrics to an algebro-geometric notion called K-stability. I will then discuss an extension of these ideas to the setting of maps between complex manifolds. On the algebraic notion this leads to a notion of stability which generalises Kontsevich's definition for Riemann surfaces, while on the analytic side the canonical Kähler metrics are analogous to metrics with prescribed curvature on Riemann surfaces. This is joint work with Julius Ross.