10 problems

Let $(C,S)$ be a pair consisting of a nonsingular projective curve $C$ and a finite set $S\subset C$, and let $h(\nu)$ be a fixed Hilbert polynomial. Fix a type $K$ of smooth proje…

Let $C'$ be a semistable curve over $F$, and let $C_L$ be the model associated with a finite nonramified extension $L$ of $F(C')$. Geometric Shafarevich conjecture. There is a numb…

Let $K(C)$ be a field as above, and let $\rho$ be an ideal of the field $F^{\mathrm{nr}}$. Strong arithmetic Shafarevich conjecture. There exists a finite extension $F$ of $K$ and…

Let $K$ be a number field, let $S$ be a finite set of places of $K$ containing all archimedean places, and let $g>0$. Denote by $\mathcal{M}_g$ the moduli stack of principally pola…

McMullen's geometric Shafarevich conjecture. For a given base $C$, there are only finitely many Kodaira fibrations $f:S\to C$ with fiber genus $g(S_b)=g$. This finiteness statement…

Fix a number field $K$ and a finite set of places $S$. A smooth projective $K$-variety is hyper-Kähler if the associated complex variety is hyper-Kähler. Unpolarized Shafarevich co…

Pointed Shafarevich conjecture. The set of morphisms $f:C\to U$ satisfying $f(c)=u$ is finite. This predicts finiteness of families over a curve after fixing one fibre over one mar…

Let $F$ be a number field, let $G$ be a connected reductive group over $F$, let $S$ be a finite set of places containing the ramified finite places of $G$ and all infinite places,…

Let $K=K^lK_l$ be a neat compact open subgroup of $G(\mathbb A_f)$, let $x=[s,\overline{1}]$ be a point of $Sh_K(G,X)(L)$, and fix a lift $\widetilde{x}=[s,1]$ in $Sh_{K^l}(G,X)$.…

Higher-dimensional Shafarevich conjecture. (B) Families of canonically polarized manifolds over $U$ with Hilbert polynomial $h$ fall into finitely many deformation equivalence clas…