80 problems

Ambro's lower semi-continuity conjecture. This function is lower semi-continuous. The conjecture concerns variation of minimal log discrepancies in families of singularities and, t…

Let $X$ be a fixed normal variety of dimension $n$, and let the point vary among its closed points. The minimal log discrepancy (mld) assigns an invariant to each such point. Lower…

Fix a dimension, and let the minimal log discrepancy be the invariant denoted by $\operatorname{mld}$ for singularities in that dimension. Shokurov's conjecture. The minimal log di…

Mustăță–Nakamura conjecture. Fix $N$ and the exponent $e$ of real ideals. Then, there exists a number $\ell_{N,e}\in{\mathbb N}$ depending only on $N$ and $e$ such that for any rea…

Let $P\in X$ be the germ of a smooth threefold, let $\mathfrak{m}$ be the maximal ideal in $\mathscr{O}_X$ defining $P$, and let … be $\mathbf{R}$-ideals on $X$. Fix a finite subse…

Shokurov's index conjecture. There exists a positive integer $r(n,a)$ such that, whenever $\operatorname{mld}_p(X)=a$, the Cartier index of $K_X$ at $p$ is at most $r(n,a)$.

Let $P\in X$ be the germ of a normal variety, and let $S+B$ be an effective $\mathbf{R}$-divisor on $X$ such that $S$ is a normal prime divisor not appearing in $B$. Assume that…

Let $(X,Y)$ be a pair, where $Y$ is a formal rational or real linear combination of closed subvarieties of $X$. For a log resolution $f:X'\xrightarrow{\ \ } X$, write … and define…

Let $(X, \mathfrak{a})$ be a log pair over an algebraically closed field $k$ of characteristic zero, and let $D$ be a normal Cartier prime divisor on $X$. Let $x \in D$ be a closed…

Shokurov's index conjecture. Klt singularities with the same mld should have similar indices of their canonical divisor.

Let $(X,\mathfrak{a})$ be a log pair, and let $|X|$ denote the set of closed points of $X$ with the Zariski topology. Write $\operatorname{mld}_x(X,\mathfrak{a})$ for the minimal l…

Let $X$ be a normal $\mathbb{Q}$-Gorenstein variety, and let $\Delta$ be an $\mathbb{R}$-Weil divisor on $X$ such that $K_X+\Delta$ is $\mathbb{R}$-Cartier. For each $d$, let…

Shokurov's codimension bound conjecture. The following inequality holds:

The precise inverse of adjunction conjecture.

Let $(X,B)$ be a log variety, and let $a(x)=a(x;X,B)$ denote the minimal log discrepancy at a closed point $x\in X$. Lower semicontinuity conjecture. For every closed point…

Shokurov's conjecture. The following properties hold: (1) $\operatorname{Mld}(d,{\mathcal B})$ satisfies the ascending chain condition; (2) $a(P;X,B)\leq \dim(X)$. Moreover, if…

Let $d$ be a positive integer and let $(X,B)$ be a log canonical pair of dimension $d$, with boundary coefficients in a set $[0,1]$ satisfying the descending chain condition. For a…

Let $x\in X$ be a point on a variety with log canonical singularities. Denote $e:=\dim m_x/m_x^2$ and $d:=\dim X$ at $x$. The minimal log discrepancy at $x$ is denoted by…

Let $X$ be a normal $Q$-Gorenstein variety, and let $xin X$ be a closed point. The Shokurov boundedness conjecture. One has … The conjecture asserts a dimension-only upper bound f…

Smallest minimal log discrepancy conjecture. For any even (respectively, odd) integer $n\geq 2$, the smallest minimal log discrepancy of an exceptional Fano variety of dimension…

Let $X$ be a klt variety, let $x \in X$ be a closed point, and let $\mathfrak{m}_x$ denote the maximal ideal corresponding to $x$. Ideal-adic semi-continuity conjecture. For every…

Let $Y$ be a normal $\mathbb{Q}$-Gorenstein variety and let $\mathfrak{a}$ be a non-zero $\mathbb{R}$-ideal sheaf on $Y$. LSC conjecture. The function … is lower semi-continuous, w…

Let $\varphi: Y \to T$ be a flat morphism, let $\mathfrak{a}$ be an $\mathbb{R}$-ideal sheaf on $Y$, and let $\tau: T \to Y$ be a section of $\varphi$. For a closed point $t \in T$…

PIA conjecture. One has

For each dimension $n\geq 2$, consider Esser's klt Calabi–Yau variety constructed in the paper, whose minimal log discrepancy is $1/m_n$. The minimal log discrepancy (mld) is the i…