37 problems

Bounded-volume boundedness conjecture. If

Let $(X,\Delta;x)$ be a klt singularity, and let $\operatorname{\widehat{vol}}(X,Delta;x)$ denote its normalized volume. Say that a collection of singularities is bounded up to spe…

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.

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 $(X,\Delta)$ be a KLT log Fano pair over $\mathbb{C}$. Let $\mathscr{s}_{-}(X,\Delta)$ denote its lim-inf $F$-signature over sufficiently large characteristic reductions, and l…

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…

Second-largest local volume conjecture. The second largest number in $\widehat{\rm Vol}_{n}$ is

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…

Let $(X,D)$ be a projective klt pair with absolute coregularity $c$. The orbifold fundamental group of the regular locus, denoted by $\pi_1^{\rm reg}(X,D)$, is taken for the orbifo…

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

Let $X$ be a compact Kähler normal space with klt singularities, and let $\mathcal{F}$ be a stable sheaf on $X$. The Bogomolov-type inequality established in the three-dimensional…

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…

Extremal del Pezzo conjecture. The klt del Pezzo surface

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 $(R,d53f)$ 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…

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.

Let the moduli space parametrize canonically polarized projective varieties with klt singularities, and let its compactification be the compactified KSB moduli space. A Weil–Peters…

Let $x\in X$ be a complex klt singularity of dimension $n$, and let $\pi_1^{\mathrm{loc}}(X,x)$ denote its local fundamental group. The Jordan property for a class of groups means…

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…