22 problems

Let $(R,\mathfrak{m})$ be a KLT singularity essentially of finite type over $\mathbb{C}$. For every sufficiently large prime $p$, let $\mathscr{s}_p(R)$ denote the $F$-signature of…

Hara–Watanabe conjecture. $(X,\Delta)$ is of strongly $F$-regular type if and only if $(X,\Delta)$ is of klt type.

ODP conjecture.

Let $(R,\mathfrak m)$ be a $d$-dimensional Gorenstein local domain essentially of finite type over $\mathbb{C}$. Gorenstein positivity conjecture. If $\operatorname{Spec} R$ has kl…

Let $R=\left(\mathbb{C}[X_1, \dots, X_n]/(f_1, \dots, f_r)\right)_{(X_1, \dots, X_n)}$, where each $f_i\in\mathbb{Z}[X_1,\dots,X_n]$, and let $R\bmod p$ denote its reduction modulo…

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 $d$ and $c$ be the dimension and absolute coregularity, respectively, and let $(X,D)$ be a klt Calabi–Yau type pair of dimension $d$ and absolute coregularity $c$. Write…

Bounded-volume boundedness conjecture. If

Boundedness conjecture. The family $\mathcal{S}_{klt}(d,\Phi,\leq u,v)$ is bounded.

Let $d$ be a positive integer and let $\Lambda$ be a set of rational numbers satisfying the descending chain condition. A projective $d$-dimensional klt log Calabi–Yau pair is a pr…

Minimal-volume conjecture for exceptional Fano varieties. For every integer $n\geq 2$, $X_n$ has the minimum anticanonical volume among all exceptional klt Fano $n$-folds.

Let $Y$ be a projective klt variety with numerically trivial canonical divisor, meaning that $K_Y$ is numerically equivalent to zero. Its augmented irregularity is the supremum of…

Let $(X,\Delta)$ be a projective klt pair with nef anti-log canonical divisor $-(K_X+\Delta)$. Here $X_{\operatorname{reg}}$ denotes the regular locus of $X$, and…

Finite-generation conjecture. The associated graded ring $\operatorname{gr}_v R$ is finitely generated.

Kollár's local fundamental group conjecture. The group $\pi_1^{\rm loc}(X,x)$ is finite.

The klt index conjecture. There exists a positive integer $n$, depending only on $d$ and $\mathfrak{R}$, such that

The absolute index conjecture. There exists a positive integer $n$, depending only on $d$, such that

Finite-degree multiplication conjecture. The local volumes satisfy

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 $(X,D)$ be a klt pair and let $x\in X$ be a singular point, so that $\operatorname{Link}(x\in X)$ denotes the link of the singularity. Local finiteness conjecture. The local fu…

Let $0\in (X,\Delta)$ be a klt singularity. The local fundamental group is … where $L(0\in X)$ is the link of the singularity. Kollár's finiteness conjecture. The local fundamental…