9 problems

Work over an algebraically closed field $k$ of characteristic $p>0$. A pair $(X,\Delta)$ is log liftable if it admits the lifting property for the pair and its boundary divisors sp…

Miyanishi's conjecture. The smooth part $T-\operatorname{Sing} T$ is rationally connected.

Let $S$ be a rank-one log del Pezzo surface, and let $S^0$ denote its smooth locus. A finite étale cover of $S^0$ is affine ruled if it contains an open subset isomorphic to a prod…

Let $(S,C)$ be a strongly asymptotically log del Pezzo surface with $r=1$, where $S$ is a smooth surface and $C$ is a smooth irreducible curve. The notation $(K_S+C)^2$ denotes the…

Let $(X,\Delta)$ be a log del Pezzo pair over a perfect field of characteristic $p>5$, with standard coefficients. A log resolution is a resolution $f\colon Y\to X$ such that the r…

Zero-angle limit conjecture. As $\beta$ tends to zero, $(S,C,\omega_\beta)$ converges in an appropriate sense to a generalized Kähler–Einstein metric $\omega_\infty$ on…

Asymptotic logarithmic Kähler–Einstein conjecture. The surface $S$ admits conical Kähler–Einstein metrics with angle $\beta$ along $C$ for all sufficiently small $\beta$ if and onl…

Small-angle metric degeneration conjecture. As $\beta$ tends to zero, $(S,C,\omega_\beta)$ converges in an appropriate sense to a generalized Kähler–Einstein metric $\omega_\infty$…

Uniformization conjecture. The surface $S$ admits Kähler–Einstein edge metrics with angle $\beta$ along $C$ for all sufficiently small $\beta$ if and only if