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

Let $(X,L)$ be a polarised projective scheme, where $L$ is an ample line bundle, and let $c_1(L)$ denote its first Chern class. A polarised scheme is $K$-polystable when the Donaldson–Futaki invariant is nonnegative for every test configuration and vanishes only for product test configurations. A Kähler metric in $c_1(L)$ is cscK if it has constant scalar curvature.

Donaldson–Tian–Yau conjecture. $(X,L)$ admits a cscK metric in $c_1(L)$ if and only if it is $K$-polystable.

This is the general cscK analogue of the Yau–Tian–Donaldson correspondence. The conjecture is stated as well known in the source and is noted there to have been solved when $L=-K_X$, namely for Fano manifolds; the general polarised case remains open.