26 problems

Let $f:X\to B$ be a projective morphism of smooth projective varieties $X$ and $B$ with connected fibers. Let $D$ denote the divisorial part of the discriminant locus of $f$. Suppo…

Let $f^\circ:X^\circ\to S^\circ$ be a family of varieties, and let $\omega_{X^\circ/S^\circ}$ denote its relative dualizing sheaf. A line bundle is relatively semi-ample if some po…

Let $f^\circ:X^\circ\to S^\circ$ be a smooth family of canonically polarized varieties. Write $\operatorname{Var}(f^\circ)$ for its variation and $\kappa(S^\circ)$ for the logarith…

Local constancy conjecture. The pluri-genera $P_k(\mathscr{X}_s)$ are locally constant as functions of $s\in S$.

Let $k$ be an algebraically closed field of characteristic zero. Let $F:X\rightarrow B$ be a surjective morphism with geometrically integral generic fiber. An alteration is a prope…

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.

Strict Arakelov inequality. The slope of $W$ satisfies

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 of manifolds with ample canonical bundle, so the fibers are canonically polarized. Suppose that the algebraic structure of the family v…

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 $f^\circ:X^\circ\to V$ be a family of canonically polarized manifolds of maximal variation over a quasi-projective variety $V$. Shafarevich's conjecture. The base $V$ is of log…

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…

Let $f_U:U\to V$ be a smooth projective family whose geometric general fiber is canonically polarized. The family has maximal variation when its fibers vary maximally in moduli. Sh…

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…

Loughran–Sofos conjecture. If the set being counted is non-empty and the fibre over every codimension $1$ point of $\mathbb{P}^n$ contains an irreducible component of multiplicity…

Let $f:X\to B$ be a smooth algebraic family of canonically polarized manifolds over a smooth quasiprojective base $B$, and let $Var(f)$ denote the number of moduli of the family. T…

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…