39 problems

Let … be a $\mathbb{Q}$-Gorenstein flat family of klt singularities with a section $t\in T\mapsto x_t\in\mathcal{X}_t$. Constructibility conjecture. The function … is constructible…

ODP conjecture.

Let $y\in Y$ be an $n$-dimensional klt singularity that is not smooth. The Gap Conjecture. normalized volume satisfies … and equality holds if and only if $y\in Y$ is an ordinary d…

Let $(X,D,x)$ be a germ of log canonical surface singularity with $\mathbb{Q}$-boundary. Borbon–Spotti conjecture. … The conjecture relates Langer's local orbifold Euler number to…

Let $\mathbbm{k}$ be a field of characteristic zero, let $x\in(X,D)$ be a klt singularity over $\mathbbm{k}$, and let $\bar{\mathbbm{k}}$ be an algebraic closure. A valuation…

Li's normalized-volume conjecture for the exceptional weighted valuation. For all other cases of $(n,k)$, ${\widehat{\rm vol}}$ should be minimized at the valuation associated to t…

Li's normalized-volume minimization conjecture. The normalized volume should have a global minimizer on ${\rm Val}_{Y,o}$ for every ${\mathbb Q}$-Gorenstein klt singularity $(Y,o)$…

Let $x\in X$ be an $n$-dimensional klt singularity, and let $\widehat{\operatorname{vol}}(x,X)$ denote its local volume, defined as the infimum of the normalized volumes of valuati…

Let $f\colon\big(y\in(Y,\Delta_Y)\big)\to\big(x\in(X,\Delta)\big)$ be a finite surjective morphism between klt pairs, and suppose it is crepant, meaning … Let…

Let $x\in X$ be an $n$-dimensional klt singularity in characteristic $0$, and for primes $p\gg0$ let $x_p\in X_p$ be its reduction modulo $p$. Let $s(x_p,X_p)$ denote the F-signatu…

Let $n$ be a positive integer, let $\varepsilon>0$, and let $\mathrm{mld}^{\mathrm{K}}(x,X)$ denote the K-minimal log discrepancy of an $n$-dimensional klt singularity $x\in X$. K-…

Let $n$ be a positive integer, and let a smooth point mean a nonsingular $n$-dimensional klt singularity. Local volume gap conjecture. There exists $\varepsilon>0$ such that the on…

Let $n$ be a positive integer, and let an ordinary double point denote the corresponding $n$-dimensional klt quadratic singularity. Ordinary double point volume gap conjecture. The…

Let $n$ be a positive integer. ACC conjecture. The set of all possible local volumes $\widehat{\rm vol}(x,X)$ of $n$-dimensional klt singularities is discrete away from zero. This…

Let $n$ be a positive integer and let $\varepsilon>0$. A $\delta$-plt blowup of a klt singularity $x\in X$ is a plt blowup $Y\to X$ of a Kollár component $E$ such that $(Y,E)$ is…

Let $n$ be a fixed positive integer. The local volume $\widehat{\rm vol}(x,X)$ is the normalized-volume invariant of an $n$-dimensional klt singularity $x\in X$. ACC conjecture for…

Let $n$ be a positive integer, let $\varepsilon>0$, and let a set of klt singularities be bounded up to special degenerations if every member admits a special degeneration to a mem…

Let $x\in X=\mathbf{Spec}(R)$ be a klt singularity, let $\mathrm{Val}_{X,x}$ denote the valuations centered at $x$, let $\widehat{\rm vol}$ be the normalized volume, and let…

Let $(R,d53f)$ be a klt singularity, and consider the normalized volume function on valuations centered at $d53f$. A valuation is quasi-monomial if it is monomial in suitable local…

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

Let $x\in X$ be a klt singularity. For a valuation $v$ centered at $x$, let $\widehat{\operatorname{vol}}_{X,x}(v)$ denote the normalized volume, and call a valuation minimizing th…

Normalized-volume conjecture. The analogous statements to the transformation theorem and the bound for the local étale fundamental group should hold with normalized volume in place…

Let $d\geq2$ be an integer and let $\Gamma\subset[0,1]$ be a DCC, respectively finite, set. For a klt germ $(X\ni x,B)$, write $\widehat{\operatorname{vol}}(X\ni x,B)$ for its norm…

Let $x\in (X,\Delta)$ be a klt singularity over characteristic $0$, and let $(x_p\in (X_p,\Delta_p))$ be its reduction modulo $p\gg 0$. Reduction conjecture. … This conjecture prop…

Let $x\in X$ be an $n$-dimensional singular point. Uniform bound conjecture. … The paper attributes this conjecture to Sotirov and Spotti and notes related work by Liu and Xu.