12 problems

Let $\mathcal{G}$ be a log bounded set of couples that is complete under dominant small modifications, and let $\widehat{\mathcal{G}}$ be its K-moduli completion: a couple belongs…

Let $(X,\Delta)$ be a KLT log Fano pair over $\mathbb{C}$, and suppose $(X_A,\Delta_A)$ is a family of characteristic-$p$ models of $(X,\Delta)$ for sufficiently large $p$. Uniform…

Let $(X,\Delta)$ be a KLT log Fano pair over $\mathbb{C}$. Let $\mathscr{s}_{-}(X,\Delta)$ denote its lim-inf $F$-signature over sufficiently large characteristic reductions, and l…

Let $(X,\Delta)$ be a projective log Fano pair over $\mathbb{C}$, and let $(X_p,\Delta_p)$ denote reductions to characteristic $p$ for sufficiently large primes. Let…

Let $n$ be a positive integer, and let $(X,\Delta)$ be a log canonical Fano pair. Write $\pi_1^{\rm orb}(X,\Delta)$ for its orbifold fundamental group. Jordan conjecture for log ca…

Let $(X,\Delta)$ be a log Fano pair defined over a field $k$, not necessarily algebraically closed, and consider test configurations of $(X,\Delta)$ that are defined over $k$. Galo…

Let $(X,\Delta)$ be a complex Fano variety or, more generally, a log Fano pair equipped with a group action. Equivariant K-stability conjecture. The pair is K-semistable, respectiv…

Let $(X,D)$ be a complex log Fano pair equipped with an action by a reductive group $G$. A $G$-equivariant test configuration is a test configuration compatible with this group act…

Structure conjecture for sRC quotients. There exists an orbifold morphism

Anticanonical Boundary Conjecture. Suppose $X$ is a klt Fano variety of dimension $n$ with $\delta(X)<1$. Then there is a general $D\in |-K_X|_{\mathbb Q}$ such that…

Let $(X,\Delta)$ be a log Fano pair, meaning that $(X,\Delta)$ is klt and $-(K_X+\Delta)$ is ample, and let $\delta(X,\Delta)$ be its stability threshold. A $\mathbb{Q}$-divisor…

Let $(X,D)$ be a log Fano pair, meaning that $(X,D)$ has klt singularities and $-(K_X+D)$ is ample. Assume … Let $X^0\subset X$ be an open subset with…