The Tian–Yau–Donaldson conjecture on K-stability and Kähler–Einstein metrics

Let $M$ be a Fano manifold, meaning a compact complex manifold with positive first Chern class. A Fano manifold is K-stable when it satisfies the algebro-geometric K-stability condition for its test configurations. Tian–Yau–Donaldson conjecture. A Fano manifold admits a Kähler–Einstein metric if and only if it is K-stable.

This conjecture links the existence of canonical metrics to algebraic stability. It has been proved in the appropriate general form: a Fano manifold admits a Kähler–Einstein metric exactly when it is K-polystable, with the formulation using K-stability depending on the convention and on whether automorphisms are present.