Generalized Yau–Tian–Donaldson conjecture for extremal metrics

Let $M$ be a compact polarised manifold and let $L$ be an ample line bundle on $M$. An extremal metric in the class $c_1(L)$ is a Kähler metric whose scalar curvature has holomorphic gradient vector field. Let a pair $(M,L)$ be K-polystable relative to a maximal torus of automorphisms when it satisfies the corresponding relative K-polystability condition. Generalized Yau–Tian–Donaldson conjecture. The manifold $M$ admits an extremal metric in the class $c_1(L)$ if and only if the pair $(M,L)$ is K-polystable relative to a maximal torus of automorphisms of $(M,L)$. This extends the expected correspondence between canonical metrics and algebraic stability from cscK metrics to extremal metrics. The paper proves a stability implication for extremal metrics, while the converse remains open in the general setting.