A big problem in Riemannian geometry is the search for a "best possible" Riemannian metric on a given compact smooth manifold. When the manifold is complex, one very nice metric we could look for is a Kahler-Einstein metric. For compact Kahler manifolds with non-positive first chern class, these were proven to always exist by Aubin and Yau in the 70's. However, the case of positive first chern class is much more delicate, and there are non-trivial obstructions to existence. It wasn't until this decade that a complete abstract characterisation of Kahler-Einstein metrics became available, in the form of K-stability. This is a purely algebro-geometric stability condition, whose equivalence to the existence of a Kahler-Einstein metric in the Fano case is analogous to the Hitchin-Kobayashi correspondence for vector bundles. In this talk, I will cover the definition of K-stability, its relation to Kahler-Einstein metrics, and (time permitting) give some examples of how K-stability is verified or disproved in practice.
