8 problems

Toric Fano extremal metric conjecture. Every toric Fano manifold admits an extremal Kähler metric in its first Chern class.

Let $(X,L)$ be a smooth toric variety with momentum polytope $\Delta$, and let $Z\subset X$ be the divisor corresponding to the preimage of $F=\bigcup_iF_i$, a union of facets of…

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…

For a toric variety with corresponding Delzant polytope, a Delzant polytope is the integral simple polytope encoding the polarized toric variety. A sub-polytope is called semistabl…

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

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…

Compatibility conjecture. Every such extremal Kähler metric $\omega_t$ is compatible with the bundle structure.