25 problems

Let $A$ be an abelian variety over an algebraically closed globally valued field $K$. If $K$ is non-archimedean, let $k$ be its constant field and let $(A^{K/k},\operatorname{tr})$…

Let $X$ be a smooth projective surface over a field $k$, let $Y$ be a smooth projective curve over $k$, and let $f:X\to Y$ be a semistable fibration whose smooth generic fiber…

Let $K$ be a function field or a number field, let $C$ be a smooth projective curve of genus $g\geq 2$ over $\overline{K}$, let $J_C$ be its Jacobian, and fix a degree-one divisor…

Bogomolov conjecture. If $X_n$ is Zariski dense in $X$ for every $n\geq 1$, then $X$ is a torsion subvariety of $\mathbb{G}_a^g$. This is presented as a consequence of the proposed…

Positselski's pro-$\ell$ Bogomolov conjecture. The maximal pro-$\ell$ Galois group of $\sqrt[\ell^\infty]{\mathbb{K}}$ is a free pro-$\ell$ group.

Let $K\subset\overline{\mathbb{Q}}\subset\mathbb{C}$ be a number field, let $S$ be an irreducible algebraic variety over $K$, and let $\pi:A\rightarrow S$ be a family of abelian va…

Let $(X,f,\mathscr L)$ be a polarized dynamical system defined over a number field, and let $Y$ be a subvariety of $X$. Let the maximal preperiodic subvarieties contained in $Y$ be…

Poonen's Mordell–Lang–Bogomolov conjecture. For $i \geq 1$, suppose

Generalized Bogomolov conjecture. There exists $epsilon>0$ such that

Let $X\subset \mathbf{G}_m^N(\overline{\mathbf{Q}})$ be a variety, and let $V_X$ be the smallest translate of a subtorus containing $X$. Write $\widehat{\mu}^{\mathrm{ess}}(X)$ for…

Let $G$ be a semiabelian variety over an algebraically closed globally valued field $K$, with abelian quotient $\pi:G\to A$. If $K$ is non-archimedean, let $k$ be its constant fiel…

Relative dynamical Bogomolov conjecture. There exists $\epsilon=\epsilon(\mathbf{F},\mathbf{V})>0$ such that

Let $k$ be either $\mathbb{Z}$ or a field, let $K$ be a global field over $k$, let $S$ be a quasi-projective normal variety over $K$, and let $\psi:A\to S$ be an abelian scheme of…

Let $C$ be a smooth projective curve of genus $g>1$ over $\overline{\mathbb{Q}}$, and let $\alpha$ be a divisor on $C$ of degree $1$. Ullmo's Bogomolov conjecture. There is a const…

Let $X\subset G$ be irreducible and not a translate, and let $H$ be the irreducible translate of minimal dimension containing $X$. Write $\operatorname{codim}_H X$ for the codimens…

Let $A$ be an abelian variety over $\overline{K}$ and let $X$ be a closed subvariety of $A$. The geometric Bogomolov assertion. If $X$ has dense small points, then $X$ is a special…

Let $K$ be a function field with algebraic closure $\overline{K}$, let $A$ be an abelian variety over $\overline{K}$, and let $X$ be a closed subvariety of $A$. A closed subvariety…

Let $A$ be an abelian variety, let $Y\subseteq A$ be a subvariety, and let $\mu(Y)$ be its essential minimum, defined as the supremum of the real numbers $\theta$ such that … is no…

Effective Bogomolov conjecture. There is a real number $c>0$ such that every proper closed irreducible subvariety $X$ of $G$ that is not contained in a translate of a proper algebr…

Let $k$ be a number field or a function field of positive characteristic, and let $\Delta_\xi$ be the canonical Gross–Schoen cycle associated with the relevant curve or fibration.…

Let $f:X\to Y$ be as above, let $CV(f)$ denote the set of singular fibers, and for $y\in CV(f)$ let $\Gamma_y$ be the polarized metrized graph of the fiber. Write…

Let $X$ be a smooth projective surface over a field $k$, let $Y$ be a smooth projective curve over $k$, and let $f:X\to Y$ be a semistable fibration whose smooth generic fiber…

Let $(X,f,\mathscr L)$ be a polarized dynamical system defined over a number field, and let $Y$ be a subvariety of $X$ with canonical subvariety height $\hat h_{\mathscr L}(Y)$. A…

Let $(A,\mathcal{L})$ be a polarized abelian variety of dimension $g$ defined over a number field $\mathbb{K}$, and let $X\subset A$ be a transverse subvariety with finite stabiliz…

Folklore conjecture. Every abelian variety $A$ defined over a number field $K$ admits, after a finite extension of $K$, an integral model $\mathcal{A}$ satisfying H1.