Tian–Yau–Donaldson conjecture for constant scalar curvature Kähler metrics

Let $(X,J)$ be a polarized manifold, with $L$ an ample line bundle on $(X,J)$. A polarization $(X,J,L)$ is K-stable in the sense of algebraic K-stability, and $c_1(L)$ denotes the first Chern class of $L$. Tian–Yau–Donaldson conjecture. There is a smooth constant scalar curvature Kähler metric in $c_1(L)$ on $(X,J)$ if and only if $(X,J,L)$ is K-stable. This is the central existence conjecture for constant scalar curvature Kähler metrics. Its precise formulation and resolution depend on the chosen notion of stability; the source presents it as a folklore conjecture.