Yau–Tian–Donaldson conjecture for K-polystability

Let $(X_{\mathbb C},L_{\mathbb C})$ be a polarized smooth complex projective variety. A test configuration and K-polystability are understood in the Donaldson–Futaki sense: the Donaldson–Futaki invariant is positive except for product-type test configurations, equivalently those arising from an $X_{\mathbb C}$-fiber bundle over $\mathbb P^1$. A constant-scalar-curvature Kähler metric in the class $c_1(L_{\mathbb C})$ means a Kähler metric whose Kähler form represents that first Chern class and has constant scalar curvature. The Yau–Tian–Donaldson conjecture. The variety $(X_{\mathbb C},L_{\mathbb C})$ is K-polystable if and only if $X_{\mathbb C}$ admits a constant-scalar-curvature Kähler metric in the Kähler class $c_1(L_{\mathbb C})$. This is the central proposed correspondence between algebro-geometric K-polystability and canonical metrics in complex differential geometry; the source presents it as a conjecture and provides no resolution status.