17 problems

Let $f^\circ:X^\circ\to S^\circ$ be a smooth family of canonically polarized varieties. The family has maximal variation when its variation equals the dimension of the base. Viehwe…

Semistable reduction conjecture. There is a projective alteration $B_1\to B$ and a projective modification $Y\to X\times_B B_1$ such that

Let $f\colon (\mathcal{X},\mathcal{L})\to S$ be a flat family of $n$-dimensional polarized smooth Calabi–Yau manifolds over a quasiprojective curve $S$. Let $\overline{S}$ be a pro…

Let $f\colon \mathcal{X}\dashrightarrow \mathcal{X}$ be a bimeromorphic map of a family of polarized abelian varieties that fixes the $0$-section. A regularizability conjecture ass…

Relative Green-Griffiths-Lang conjecture. The following conditions are equivalent:

General-fiber quasi-$F$-splitting conjecture. If the relative canonical divisor is trivial and $X$ is quasi-$F$-split, then a general fiber of $\pi$ is quasi-$F$-split.

Arakelov equality characterization. The general fiber has Kodaira dimension zero, and the family is Shimura in the following sense: $Y$ parameterizes a compactified universal famil…

Let $f:X\to Y$ be a smooth projective family between complex quasi-projective manifolds. Assume that each fiber of $f$ has semi-ample canonical bundle, and that the Kobayashi pseud…

Let $X$ be the variety occurring in the family $\Delta^k(X)$ over $(\mathbb{C}^k)^{\circ}$. For $(m_1,\ldots,m_k),(n_1,\ldots,n_k)\in(\mathbb{C}^k)^{\circ}$, define an equivalence…

Let $f_U:U\to V$ be a smooth family of projective varieties, and let $(X,D)$ be a smooth compactification of $V$ with $V\cong X\smallsetminus D$. Assume that the geometric general…

Let $f_U:U\to V$ be a smooth projective family whose general fiber admits a good minimal model. Write $\operatorname{Var}(f_U)$ for the variation of the family. Kebekus–Kovács hype…

F-rational fiber-locus conjecture. The locus of $t\in\mathfrak{mSpec}D$ for which $X_t$ is $F$-rational is constructible. If the family is flat and proper, or sufficiently local, t…

Let $X^{\circ}$ be a smooth quasi-projective variety, and let $f^{\circ}:Y^{\circ}\to X^{\circ}$ be a smooth family of canonically polarized varieties with maximal variation. Write…

Let $F$ be a number field, let $X$ satisfy the stated assumptions, and let $H$ be an anticanonical height function on $X$. Let $\pi:Y\to X$ be a non-singular proper faithfully flat…

Let $X$ satisfy the stated assumptions, let $H$ be an anticanonical height function, and let $\pi:Y\to X$ be a proper surjective almost smooth morphism with geometrically integral…

Let $\pi: \mathscr{S} \to B$ be a family of surfaces over an analytic space $B$, and call a point of $B$ very general if it lies outside a countable union of proper analytic subset…

Let $f:X\to B$ be an algebraic smooth family of canonically polarised manifolds with generically maximal variation, and let $B$ be a special quasi-projective manifold, meaning that…