Abstract

Much of Riemannian geometry and geometric analysis centres on finding a ``best possible” metric on a fixed smooth compact manifold. One very nice metric on a compact complex manifold that we could ask for is a Kahler-Einstien metric, the study of which goes back to the 50’s with the Calabi conjecture. 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 an 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 (and more generally extremal) metrics, and give some examples of how K-stability is calculated in practice.