Extremal metric–relative K-stability conjecture

Let $(X,L)$ be a polarised variety, let $T$ be a maximal torus of automorphisms of $X$, and let $F_\chi(\alpha)$ denote the modified Futaki invariant associated with the extremal vector field, for test-configurations commuting with $T$. The variety is K-stable relative to $T$ when $F_\chi(\alpha)$ is non-negative for all such test-configurations, and vanishes only for test-configurations arising from a $\mathbf{C}^*$-action on $X$. Extremal metric–relative K-stability conjecture. A polarised variety admits an extremal metric in the class of the polarisation if and only if it is K-stable relative to a maximal torus. This conjecture proposes the stability criterion for the existence of extremal Kähler metrics; the source introduces it as a modification of K-stability designed to account for automorphisms, and does not give a resolution.