27 problems

Let $A$ be the coefficient ring in the reduction-modulo-$p$ setup, let $\frak{a}\subset A[x_1,\ldots,x_n]$ be an ideal, let $c$ be its log canonical threshold, and let $c_p$ denote…

Kawamata's conjecture in the Fano case. One has

Let $X$ be a weakly pseudoconvex manifold, let $\mathscr{J}$ be an ideal sheaf on $X$, and let $L\longrightarrow X$ be a holomorphic line bundle. Assume that $L$ is exhaustion sing…

Jonsson–Mustață's equivalent conjecture. There is a quasimonomial valuation $\nu$ on $\mathcal{O}_o$ that computes $\mathrm{lct}^{\mathfrak{q}}(\mathfrak{b}_{\bullet})$, namely

Vojta's multiplier-ideal conjecture. For every real $\varepsilon>0$ and positive integer $r$, there is a proper Zariski-closed subset $Z\subsetneq\mathcal X$, depending only on…

Conjecture 1.1.3. Under the curvature assumption given in Theorem, there exists a sufficiently large constant $\ell\geq e$, depending only on $\psi$ and $\delta$, such that for eve…

Let $X$ be a smooth, irreducible complex algebraic variety and let $f\in\mathcal{O}_X(X)$ be nonzero. Choose a model $(X_A,f_A)$ over a finitely generated ${\mathbf Z}$-subalgebra…

Let $\varphi$ be plurisubharmonic in a bounded pseudoconvex neighborhood of $0\in\mathbb{C}^n$, locally bounded away from $0$. For $m\in\mathbb{Z}_{>0}$, let $\varphi_m$ be the ana…

Demailly's conjecture. The $n$-th Lelong numbers converge:

Maximal non-lc reduction conjecture. The reduction of the maximal non-lc ideal filtration coincides with the non-$F$-pure ideal filtration on a dense set of closed fibers, as asser…

Let $X$ be a $g$-dimensional abelian variety and let $B$ be an ample rational divisor on $X$. The cosupport of a multiplier ideal is the locus where that ideal is nontrivial. Multi…

Let $(A,\fra^t)$ be a pair, where $A$ is the regular ring and $\fra$ is the ideal appearing in the construction of the perfectoid test ideal ${\mathrlap{\mJ}\,\,\uptau}(A,\fra^t)$.…

Let $X$ be a complex normal projective variety and $D$ a Weil divisor on $X$. Global generation conjecture. There is an ample Cartier divisor $H$ on $X$ such that for any…

Let $X$ be a normal variety over a field $k$ of characteristic zero, let $\Delta$ be a $\mathbb{Q}$-divisor such that $K_X+\Delta$ is $\mathbb{Q}$-Cartier, and let $\mathfrak{a}$ b…

Let $F=(f_1,\ldots,f_r)$ be a collection of non-zero polynomials $f_j$ in $\mathbb{C}[x_1,\ldots,x_n]$, and let $B_F\subset\mathbb{C}[s_1,\ldots,s_r]$ be its Bernstein-Sato ideal,…

Demailly–Kollár jumping-number openness conjecture. If $c_x^{\mathfrak q}(\varphi)<\infty$, then $|\mathfrak q|^2\exp(-2c_x^{\mathfrak q}(\varphi)\varphi)$ is not locally integrabl…

Let $R$ be an excellent regular domain of equicharacteristic zero, let $\mathfrak b_\bullet$ be a subadditive sequence of ideals in $R$, and say that it has controlled growth when…

Let $R$ be an excellent regular domain of equicharacteristic zero, let $\mathfrak a_\bullet$ be a graded sequence of ideals in $R$, and let $\mathfrak q$ be a nonzero ideal in $R$.…

Let $X$ be a smooth variety, let $L$ be a pseudo-effective $\mathbb{R}$-divisor on $X$, and let $T_{\min}$ be a current of minimal singularities in the numerical class of $L$. Write…

Let $\mathfrak{a}\subset{\mathbf Z}[x_1,\ldots,x_n]$ be an ideal as above, let $\mathfrak{a}_p$ be its reduction modulo $p$, and let $\tau(\mathfrak{a}_p^\lambda)$ and…

Let $X$ be a normal variety over an algebraically closed field $K$ of characteristic zero, and let $D$ be an effective $\mathbb{Q}$-divisor on $X$ such that $K_X+D$ is $\mathbb{Q}$…

Reduction-to-characteristic-$p$ multiplier ideal conjecture. For all sufficiently large primes $p$,

Let $Y$ be a smooth, irreducible variety over an algebraically closed field $k$ of characteristic zero, and let $\mathfrak{a}$ be a nonzero ideal on $Y$. Given any model $Y_A$ and…

Quasi-monomiality conjecture. The weak version asserts that, for every generic point $\xi$ of an irreducible component of the subscheme defined by…

Strong openness conjecture. The following equality of sheaves holds: