Abstract: A fundamental problem in complex geometry is to construct canonical metrics, such as Hermite-Einstein (HE) metrics on vector bundles and constant scalar curvature Kähler (cscK) metrics on Kahler manifolds. On a given vector bundle/manifold, such a metric may or may not exist, in general. The existence question for such metrics has been found to have deep connections to algebraic geometry. In the case of vector bundles, the Hitchin-Kobayashi correspondence proved by Uhlenbeck–Yau and Donaldson show that the existence of a HE metric is captured by the notion of slope stability for the vector bundle. In the case of manifolds, the still open Yau-Tian-Donaldson conjecture relates the existence of cscK metrics to K-stability of the underlying polarised variety.
Together with Ruadhaí Dervan, I started a research programme where we study canonical metrics, called Optimal Symplectic Connections, and a notion of stability, on fibrations. We proposed a Hitchin-Kobayashi/Yau-Tian-Donaldson type conjecture in this setting as well. In the case when the fibration is the projectivisation of a vector bundle, we recover the Hermite-Einstein and slope stability notions, respectively, and as such the theory can be seen as a generalisation of the classical bundle theory to more general fibrations. There has recently been great progress on this topic both on the differential and algebraic side, through works of Hallam, McCarthy, Ortu, Hattori, Spotti and Engberg, in addition to the joint works with Dervan. The aim of this talk is to give an introduction to and overview of the status of this programme.