91 problems

Tian's strong stabilization conjecture. If $R(X,L)$ is generated by

Let $X$ be a klt variety and let $\mathfrak{a}_\bullet$ be a graded sequence of ideals on $X$ with finite log canonical threshold $\operatorname{lct}(\mathfrak{a}_\bullet)$. A valu…

Let $f$ be a polynomial over $\mathbb C$, and let $f_\mu$ denote its reduction to characteristic $p$ for a maximal ideal $\mu$ of a finitely generated subring of $\mathbb C$ contai…

Let $f$ be a non-constant polynomial, and consider the Igusa local zeta function associated to $f$. A pole has maximal possible order if its order equals the largest order allowed…

Cheltsov–Park cone conjecture. If $d\geq n$, then equality holds in this bound if and only if $V$ is a cone. This conjecture proposes a characterization of the equality case for th…

Demailly's strong openness conjecture. If

Let $f:(\mathbb C^n,0)\longrightarrow(\mathbb C,0)$ be an analytic function germ, and for each $i\in\{1,\dots,n\}$ let $\theta^{(i)}(f)$ denote the Łojasiewicz exponent of the Jaco…

Fix a positive integer $d$ and a non-negative real number $a$. For DCC sets $\Gamma_1,\Gamma_2$, define the relative threshold set using pairs $(X/Z\ni z,B)$ that are $a$-lc over…

Kollár's conjecture. The accumulation points of $\operatorname{LCT}_n(I)$ are

Global threshold conjecture. For some such divisor $D$, one has

Shokurov–Kollár conjecture. The following properties hold: (1) $\operatorname{Lct}(d,{\mathcal B})$ satisfies the ascending chain condition; (2) if…

Let $d\geq 2$ be a natural number, $a\geq 0$, let $\\{0\\}\subseteq\Gamma\subseteq[0,1]$ satisfy the dcc, and let $\\{0\\}\subsetneq S\subset\mathbb R$ be a finite set of nonnegati…

ACC for log canonical thresholds. The displayed set satisfies the ascending chain condition (ACC).

Let $\frak{a}$ be an ideal in $\mathbb{Z}[X_1,\ldots,X_n]$, contained in $(X_1,\ldots,X_n)$, and let $\frak{a}_p$ be the localization at $(X_1,\ldots,X_n)$ of its image in…

For each dimension $n$, consider the set of log canonical thresholds and its set of accumulation points. Kollár's conjecture. The set of accumulation points in dimension $n$ is equ…

Accumulation-point conjecture. The accumulation set of $\mathcal{T}_d$ is precisely

Ascending chain conjecture. The set $\mathcal{T}_d$ satisfies the ascending chain condition: every increasing chain of elements of $\mathcal{T}_d$ terminates.

Generalized Eckardt point conjecture. If the total log canonical threshold of $X$ is $\frac{n-1}{n}$, then a wild tiger $S$ for $X$ is a cone in $\mathbb{P}^{n-1}$ over a smooth hy…

McKernan's ACC conjecture. There exists an ACC set $\Gamma\subset [0,1]$ depending only on $d$ such that, for every such $\mathcal{F}$ on $X$, the threshold…

Let $d_n$ be the positive integer appearing in the Sylvester-number construction, and consider log canonical thresholds of hypersurfaces (more generally, of pairs). Kollár's Gap co…

Jonsson–Mustaţă's weighted conjecture. The weak version asserts that there exists a quasi-monomial valuation $v\in\mathrm{Val}_X^*$ computing…

Let $\mathbb{K}$ be the base field, let $\overline{G}\subset \mathrm{GL}_n(\mathbb{K})$ be a finite subgroup containing no reflections, and let $G$ be its image in…

Let $R$ be a singularity in characteristic zero and let $I$ be an ideal. For reductions modulo primes $p$, write $R/p$ for the reduction of $R$ and $IR/p$ for the induced ideal. Th…

Folklore monotonicity conjecture. There exists $k_0\in\mathbb{N}_+$ such that, whenever $k_0\leq k_1\leq k_2$,

Let $I$ be an ideal of $\mathcal O_n$. Say that $I$ is diagonal with respect to coordinates $x_1,\dots,x_n$ if it is generated, up to integral closure, by pure powers of these coor…