49 problems

Analytic version of Jonsson–Mustaţă's conjecture. If $c_o^{\mathfrak{q}}(\varphi)<+\infty$, then

Let $J_{\bullet}$ be a graded sequence of ideals on $X$, let $\mathfrak{q}$ be a nonzero ideal on $X$, and suppose that the asymptotic jumping number…

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…

Mustată–Srinivas conjecture. There are infinitely many primes $p$ such that

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…

Let $(X,\Delta)$ be a log resolution of a pair, and let $(\widetilde X,\widetilde\Delta)$ be its log canonical model. A divisor is relevant for jumping numbers if it contributes to…

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…

Mustata's conjecture. For hyperplane arrangements, the jumping coefficients depend only on the combinatorial data of the hyperplane arrangement.

Kawamata's conjecture in the Fano case. One has

Kawamata's conjecture. One has

Equality conjecture for big line bundles. If $L$ is big, then

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…

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 that is exhaustion singular-positiv…

Let $f\colon X\to Y$ be a fibration of complex normal projective varieties, let $(X,B)$ be a pair, let $L\coloneqq -K_X-B$, let $y\in Y$ be general, and assume that $Y$ is…

Let $f\colon X\to Y$ be a fibration of complex smooth projective varieties. Assume that the asymptotic multiplier ideal $\mathcal{I}(X_y,0;\lVert -K_X\rVert_y)$ is trivial for gene…

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…

Cao–Tsuji conjecture. The limit

Vanishing conjecture for Hodge ideals. The vanishing theorem for the Hodge ideals $I_k(D)$ should still hold without the global assumption.

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 ${\mathrlap{\mJ}\,\,\uptau}(A,\fra^t)$.…

Ohsawa's matrix strong openness conjecture. If

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 an $n$-dimensional nonsingular variety over an algebraically closed field $k$ of characteristic zero, and let $\mathfrak{a}$ be a nonzero ideal on $X$. For a model $X_A$…