Yau–Tian–Donaldson conjecture for cscK metrics

Let $(X,L)$ be a smooth polarized manifold, where $c_1(L)$ denotes the first Chern class of $L$, and let K-polystability mean that every test configuration has nonnegative Donaldson–Futaki invariant, with equality only for product test configurations. Yau–Tian–Donaldson conjecture. There exists a constant-scalar-curvature Kähler metric in the class $c_1(L)$ if and only if $(X,L)$ is K-polystable. This is the cscK form of the Yau–Tian–Donaldson correspondence, relating an analytic existence problem to an algebro-geometric stability condition; its effective and general form remains open.