10 problems

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 v…

Let the Kähler cone be the set of Kähler classes, and let the extremal Kähler cone be the subset consisting of classes represented by extremal Kähler metrics. Suppose the flow equa…

Let $(M,J)$ be a complex surface of positive first Chern class polarized by a Kähler class $\Omega$. The positive first Chern class convergence conjecture. There exists an initial…

Let $(X,\omega)$ be a compact Kähler manifold and let $D$ be a smooth divisor. Let $T$ be a maximal compact torus in the reduced automorphism group…

Let $(X,J)=P(E)$ be a projective bundle over a compact curve $\Sigma$ of genus at least $2$. A subbundle is stable in the usual slope-stability sense. Apostolov–Calderbank–Gauducho…

Let $X$ be a Kähler manifold with Kähler class $[\omega]$. Suppose that there exists an extremal metric $\omega_0\in[\omega]$. Let $\omega_1\in[\omega]$ be a Kähler metric invarian…

Modified uniform K-stability conjecture. The pair $(M,\omega)$ has a metric within the class $[\omega]$ that solves the Abreu equation if and only if $(\Delta,A)$ is uniformly K-st…

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 holomorph…

Let $P\subset \mathbb{R}^{n}$ be a Delzant polytope, let $A$ be a smooth function on $\overline{P}$, and let $L_A$ be the associated linear functional on convex functions. An admis…

K-stability conjecture. $M$ admits a Kähler metric with constant scalar curvature, or more generally an extremal metric, if and only if $M$ is stable in a certain sense of geometri…