26 problems

Tangent-cone product conjecture. These spaces converge in pointed Gromov–Hausdorff topology to a product of copies of $\mathbb{R}$ and $\mathbb{C}$, compact Calabi–Yau varieties, a…

Let $x\in X$ be an $n$-dimensional normal $\mathbb{Q}$-Gorenstein singularity over an algebraically closed field of characteristic zero, and suppose that $x$ is an isolated non-log…

Shokurov's conjecture. The following properties hold: (1) $\operatorname{Mld}(d,{\mathcal B})$ satisfies the ascending chain condition; (2) $a(P;X,B)\leq \dim(X)$. Moreover, if…

Finiteness conjecture. The analogous finiteness result should hold in every dimension: for each dimension, there are only finitely many types of exceptional singularities.

Let $(R,\mathfrak{m})$ be a normal local domain essentially of finite type over a field of characteristic zero. Suppose that $R$ is lim-stable, for example, semistable. Lim-stabili…

Let $R$ be a normal $\mathbf{Q}$-Gorenstein ring. The lim-perfectoid log-canonicity conjecture. If $R$ is lim-perfectoid pure, then $R$ is log canonical. The paper proves the corre…

Let $d$ be a positive integer, let $\Gamma_0\subset[0,1]\cap\mathbb{Q}$ be finite, and let $t_0\in[0,1]$ be irrational. Uniform interpolated lc rational polytope conjecture. There…

Let $s$ be a natural number and let $\epsilon$ be a positive real number. Assume $(Z,\Delta+\mathbf N)$ is an $\epsilon$-lc generalized pair of dimension $s$ with a closed point…

Virtually nilpotent kernel conjecture. The normal subgroup $N$ is virtually nilpotent of rank at most $2n+1$.

Completeness conjecture. The metric $g_{KE}$ is complete near $p$; more precisely, there is an open neighborhood $V_{q,R}$ of $p$ such that for every $x\in V_{q,R}\setminus\{p\}$ a…

Asymptotic closeness conjecture. Any complete Kähler–Einstein metrics near log canonical singularities should be asymptotically close to each other, with a local collapsing model a…

Broustet–Höring's conjecture. If $X$ is not necessarily $\mathbb{Q}$-Gorenstein, then $X$ has log canonical singularities.

Let $(X,\Delta)$ be a normal variety over a field of characteristic zero, with $\Delta$ an effective $\mathbb{Q}$-divisor and $K_X+\Delta$ $\mathbb{Q}$-Cartier. Log canonical–dense…

Locally sharply Frobenius split type conjecture. If $(X,Delta)$ is log canonical, then it is of locally sharply Frobenius split type.

Flip conjecture II. The sequence terminates after finitely many steps; equivalently, there is no infinite sequence of flips. The source explains that the minimal model program can…

Fujiki-class finite-generation conjecture. The ring $R(X,\Delta)$ is a finitely generated $\mathbb C$-algebra. The source states that this conjecture is equivalent, after taking a…

Finite-generation conjecture. The ring $R(X,\Delta)$ is a finitely generated $\mathbb C$-algebra. The surrounding discussion distinguishes this conjecture from the established klt…

Log canonical base conjecture. There is an effective $\mathbb R$-divisor $\Delta_Y$ on $Y$ such that $(Y,\Delta_Y)$ is log canonical and

The log canonical Proj conjecture. There is an effective $\mathbb Q$-divisor $\Delta_Y$ on $Y$ such that $(Y,\Delta_Y)$ is log canonical. The result is known when $K_Y$ is…

Cat property conjecture. $X$ satisfies the Cat property if and only if every effective $bQ$-divisor $D\simq-K_X$ is log canonical in codimension $1$.

Niu's generic-linkage conjecture. The variety $Y$ is also a local complete intersection with log canonical singularities.

Log canonical versus dense $F$-pure type conjecture. The pair $(Y,\Delta)$ is log canonical if and only if $(Y,\Delta)$ has dense $F$-pure type.

Let $R=k[x_0,\ldots,x_n]$, and let $I=(f_1,\ldots,f_t)$ be a homogeneous ideal generated in degrees $d_1\geq d_2\geq \cdots\geq d_t\geq 1$, of codimension $r$. Set…

Let $S=\operatorname{Spec} R$ be a regular affine scheme over $k$, and let $X\subset S$ be a subscheme defined by $I=(z_1,\ldots,z_t)$ of codimension $r$. Let…

Projective compactification conjecture. There exists a base change morphism $\vartheta:T\to B$ and a projective lc morphism $\phi:X\to T$ such that $U_T\subseteq X$ and