4 problems

Hyperkähler cone conjecture. Any conical symplectic singularity is a hyperkähler cone.

A Fano cone singularity is a $\mathbb{Q}$-Gorenstein singularity with log-terminal singularities, and the chiral ring is the coordinate ring of the moduli space of a $4d$…

Fano-cone Ricci-flatness conjecture. On a Fano cone $X$, every Kähler-Ricci shrinker is a Ricci-flat Kähler cone metric.

Fix $n$, $V$, $T$, and a Reeb vector field $\xi$, and let $M_V^{\rm ss}(\xi)$ be the good moduli space of $n$-dimensional K-semistable Fano cones of volume density $V$, with descen…