3 problems

Equivariant K-stability equivalence conjecture. $X$ is $G$-equivariantly K-semistable if and only if $X$ is K-semistable, and $X$ is $G$-equivariantly K-polystable if and only if…

Equivariant test-configuration conjecture. To test K-(poly/semi)stability of $X$, it should be enough to examine test configurations that are equivariant with respect to $G$.

Let $\pi:X\to T$ be a projective family of varieties. Assume that $T$ is normal, every fiber $X_t$ is klt, and $-K_{X/T}$ is $\mathbb{Q}$-Cartier and ample. For a geometric point…