24 problems

ODP conjecture.

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 positive integer, let $\varepsilon>0$, and let $\operatorname{mult}_xX$ denote the multiplicity and the embedded dimension of $x\in X$ its embedding dimension. Multipl…

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

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 $x\in (X,\Delta)$ 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 $(X,D)$ be a klt pair, let $\eta$ be a non-closed point, and let $Z=\overline{\{\eta\}}$ have dimension $d$. Choose a general closed point $x\in Z$, and let $n=\dim X$. Non-clo…

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…

Finite degree formula. The normalized volumes satisfy

Let $G$ be a group acting on a klt singularity $x\in (X,D)$, with $x$ fixed by the action. Let $\widehat{\operatorname{vol}}_{(X,D),x}$ denote the normalized-volume function. Group…

Let $x\in (X=\operatorname{Spec}(R),D)$ be a klt singularity, and let $v$ be a minimizer of the normalized-volume function at $x$. Finite-generation conjecture. The associated grad…

Let $(X,d_\infty)$ be a Gromov–Hausdorff limit as in the construction of Donaldson and Sun, let $x\in X$ be a point, let $C=C_xX$ be a metric tangent cone, and let $W$ be the affin…

Let $(xi i (Xi,Di))$ be klt singularities for $i=1,2$, and write $ni:=\dim Xi$. Let $pii:X1\times X2to Xi$ be the projections. Conjectural product formula. … The formula would expres…

Stable degeneration conjecture. There is a unique minimiser $v$ up to rescaling. Furthermore, $v$ is quasi-monomial, $R_0$ is finitely generated, and the induced degeneration is a…

Algebraic ODP volume gap conjecture. The infimum of the normalized volumes of valuations centered at $p$ satisfies

Let $X$ be a variety of dimension $n$, let $x\in X$ be a closed point, and let $\operatorname{Val}_{X,x}$ denote the set of real valuations on $X$ centered at $x$. For…

Uniqueness conjecture. For any $\mathbb{Q}$-Gorenstein klt singularity $(X,o)$, there exists a unique valuation minimizing $\widehat{\rm vol}$ on ${\rm Val}_{X,o}$.