6 problems

Székelyhidi's conjecture. $X$ admits an extremal Kähler metric in the class of the polarization if and only if it is relatively K-polystable.

Let $(\Delta,\mathbf L)$ be the momentum polytope of a smooth compact toric Kähler manifold $(X,\omega_0)$, let $F$ be a fixed union of facets of $\Delta$, and let $Z\subset X$ be…

Let $k$ be the extremal Kähler metric on $\mathbb{C} P^2\sharp 2\overline{\mathbb{C} P^2}$ such that $s^{-2}k$ is the Chen–LeBrun–Weber metric with Einstein constant equal to $3$,…

Donaldson's relative K-stability conjecture. There is a solution $G\in\mathcal{S}(\Delta,u)$ to the extremal Kähler equation

Convexity criterion conjecture. The following conditions should be equivalent: $\Omega$ admits an extremal Kähler metric; $\Omega$ admits a compatible extremal Kähler metric; and

Extremal metric decomposition conjecture. The projective bundle $(M,J)=P(E)$ admits an extremal Kähler metric in some Kähler class if and only if $E$ decomposes as a direct sum of…