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

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