50 problems

Fix $n\in\mathbb{N}$, $\varepsilon>0$, and let $I\subseteq [0,1]$ be a finite set. For a klt singularity $x\in (X,\Delta)$, let $\operatorname{mld}^{\mathrm{K}}(x,X,\Delta)$ denote…

Shokurov's codimension bound conjecture. The following inequality holds:

Let $(X,B)$ be a log variety, and let $a(x)=a(x;X,B)$ denote the minimal log discrepancy at a closed point $x\in X$. Lower semicontinuity conjecture. For every closed point…

Let $(X,Y)$ be a pair, where $Y$ is a formal rational or real linear combination of closed subvarieties of $X$. For a log resolution $f:X'\xrightarrow{\ \ } X$, write … and define…

Let $x\in X$ be a point on a variety with log canonical singularities. Denote $e:=\dim m_x/m_x^2$ and $d:=\dim X$ at $x$. The minimal log discrepancy at $x$ is denoted by…

Smallest minimal log discrepancy conjecture. For any even (respectively, odd) integer $n\geq 2$, the smallest minimal log discrepancy of an exceptional Fano variety of dimension…

Let $X$ be a klt variety, let $x \in X$ be a closed point, and let $\mathfrak{m}_x$ denote the maximal ideal corresponding to $x$. Ideal-adic semi-continuity conjecture. For every…

Let $\varphi: Y \to T$ be a flat morphism, let $\mathfrak{a}$ be an $\mathbb{R}$-ideal sheaf on $Y$, and let $\tau: T \to Y$ be a section of $\varphi$. For a closed point $t \in T$…

Let $(X, \mathfrak{a})$ be a log pair over an algebraically closed field $k$ of characteristic zero, and let $D$ be a normal Cartier prime divisor on $X$. Let $x \in D$ be a closed…

PIA conjecture. One has

Lower semi-continuity conjecture. The function $x\mapsto \operatorname{mld}_x(X,\mathfrak{a})$ is lower semi-continuous.

Uniform boundedness conjecture for foliated minimal log discrepancies. For every positive integer $d$, there exists a positive real number $l$ depending only on $d$ and $\Gamma$ su…

For each integer $n\geq2$, let $V/G$ be the quotient of the hypersurface constructed in the paper by the cyclic group action described there; it is a klt Calabi–Yau variety of dime…

Ambro–Shokurov's lower semicontinuity conjecture. For each $d$, the function

Ambro–Shokurov's ACC conjecture. The set $\Omega_k$ is an ACC-set: every increasing sequence $\alpha_1\leq\alpha_2\leq\cdots$ in $\Omega_k$ eventually becomes stationary.

ACC conjecture for mlds of enc pairs. If $\Gamma$ satisfies the DCC, then $\operatorname{eMLD}_{d}(\Gamma)$ satisfies the ACC. If $\Gamma$ is finite, then…

Shokurov's Gorensteinness conjecture. The following assertions hold:

Shokurov's index conjecture. There exists a positive integer $r(n,a)$ such that, whenever $\operatorname{mld}_p(X)=a$, the Cartier index of $K_X$ at $p$ is at most $r(n,a)$.

Liu's minimal mld conjecture. The number $1/(s_{n+1}-1)$ is the smallest possible minimal log discrepancy among all $n$-dimensional klt Calabi-Yau pairs with standard coefficients.

Let $d$ be a positive integer. Consider the minimal log discrepancies of all normal projective Calabi–Yau rationally connected varieties of dimension $d$. Minimal-log-discrepancy n…

Boundedness conjecture. There exists a constant $A=A(n,I)$, depending only on $n$ and $I$, such that

Boundedness conjecture. There exists a constant $A=A(n,\varepsilon,I)$ such that

Bounded discrepancy conjecture. The divisor computing the minimal log discrepancy can be chosen with discrepancy for $(X,0)$ bounded uniformly in terms of $d$ and $\Gamma$. The cla…

Let $(X,\mathfrak{a})$ be a log pair, let $D$ be a normal Cartier prime divisor, and let $x\in D$ be a closed point. Suppose that $D$ is not contained in the cosupport of the…

Let $(X,\Delta+M)/Z$ be a generalized pair. Define … where $|X|_{\mathrm{cl}}$ denotes the set of all closed points of $X$ with the Zariski topology. Lower semi-continuity conjectu…