17 problems

Let … be a $\mathbb{Q}$-Gorenstein flat family of klt singularities with a section $t\in T\mapsto x_t\in\mathcal{X}_t$. Constructibility conjecture. The function … is constructible…

Let $S$ be a complex manifold, and let $\pi:{\mathcal X}\to S$ and $\pi':{\mathcal Y}\to S$ be smooth families of compact complex manifolds. Write ${\mathcal X}_U=\pi^{-1}(U)$ and…

Let $g:\mathcal X\to\Delta$ be a flat, proper, Moishezon morphism, where $\Delta$ is the unit disc in $\mathbb C$. Assume that $X_0$ has only canonical singularities. A morphism…

Let $g:\mathcal X\to\Delta$ be a flat, proper, Moishezon morphism over the unit disc $Delta a$ in $mathbb C$. Assume that $X0$ has only canonical singularities. A morphism is **fib…

Let $f:\text{?}$ be a family, automatically flat over a smooth curve in this paper, whose fibers all have only canonical singularities. A fiber is Moishezon if it is Moishezon, and…

Let $T$ be an irreducible variety, let $X/T$ be a family of smooth irreducible projective varieties, and let $f:X/T\dashrightarrow X/T$ be a family of dominant rational maps. Write…

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 $t \in T$…

Let $\pi:\mathcal{X}\to\mathbb{D}$ be a family as in the preceding proposition, and let $(u_k)_k$ converge in the family sense to $u_0$. Denote by $\mathbf{E}_k$ and $\mathbf{E}_0$…

Let $(X_k,\omega_k)$ be fibres in a family and let $(u_k)_k$ be a sequence of fibrewise plurisubharmonic functions converging to $u_0$ in the family sense. Continuity conjecture. T…

Let $S$ be the base of the family considered in the paper, and for a subvariety $Z\subset S$ let $G_Z$ and $G_Z^{AH}$ be its generic Mumford–Tate and generic absolute Mumford–Tate…

Let $f:X\to Z$ be as in the setting, with fibers $X_t$, forms $\theta_t$ and $\omega_t$, and let $V$ denote the common volume appearing in the fiberwise averages. Write…

Let $X$ and $S$ be quasi-projective varieties over a number field $K$, and let $\pi:X\to S$ be a projective morphism. Let $f:X\to X$ be a surjective morphism satisfying…

Let $f\colon (U,D)\to V$ be a log smooth family with $V$ quasi-projective, with all coefficients of $D$ in $(0,1)$, and suppose that every fiber $(U_y,D_y)$ is of log general type.…

Let a non-groupless projective variety be one admitting a nonconstant morphism from an abelian variety. Green–Griffiths–Lang specialization conjecture. Every specialization of a no…

Let $F$ be a number field, let $G$ be a reductive group, and let $\{\mathcal F_k\}_{k\geqslant1}$ be a “general” sequence of finite families of automorphic representations of…

Variational Hodge conjecture for de Rham cohomology. For every complex point $s$ of $S$, the class $\widetilde{\beta}_s$ is the cohomology class of an algebraic cycle.

Variational Hodge conjecture. For every complex point $s$ of $S$, the class $\widetilde{\alpha}_s$ is the cohomology class of an algebraic cycle.