17 problems

Let $X$ be a smooth irreducible $k$-variety, let $\overline{X}$ be a partial compactification of $X$, and let $\mathcal{E}$ be an $F$-isocrystal on $X$ overconvergent along…

The $p$-adic weight-monodromy conjecture. The filtrations satisfy

Rapoport–Zink spectral sequence conjecture. For all $r,w$, the induced map on $E_2$-terms is an isomorphism

Let $f:\Delta_X\rightarrow\Delta_B$ be a surjective morphism of rational conical polyhedral complexes such that $f^{-1}(0)=\{0\}$. A projective subdivision replaces the complexes b…

Polyhedral semistable reduction conjecture. There exists a projective alteration $\Delta_B^1\to\Delta_B$, with induced alteration $\Delta_X^1\to\Delta_X$, and a projective subdivis…

Toroidal lifting conjecture. There exist modifications $X'\to X$ and $B'\to B$, each a composition of blowings up with smooth centers lying over $\Sigma$, and a lifting $X'\to B'$…

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 $K$ be a complete discretely valued field. An extension $L/K$ is weakly totally ramified if it is separable and has the relevant ramification property while preserving the resi…

Log-smooth modification conjecture. There exists a finite extension $l/k$ and an admissible blowing up

Compact covering conjecture. There exists a finite extension $l/k$ and a finite covering

Let $K$ be the coefficient field, let $m$ be a positive integer, and let $\epsilon\in(0,1)$. Let $A_K[\epsilon,1)^m$ denote the corresponding polyannulus, and let $\mathcal{E}$ be…

Let $f\colon X \to B$ be a map of conical complexes. A map of conical complexes is semistable if $B$ is regular, every cone maps into a cone of $B$, the induced map on the relevant…

Let $f\colon X\rightarrow B$ be a dominant morphism of finite type of integral qe schemes. A morphism is semistable when, in the characteristic-zero setting, its source and target…

Polystable reduction conjecture. There exists a modification $Y\to X$ such that $Y_\eta=X_\eta$ and $Y$ is polystable over ${\mathcal O}$.

Let $K$ be the underlying $p$-adic field, let $F$ be the coefficient field of Hyodo–Kato cohomology, and let $X$ be a proper, log-smooth, fine and saturated…

Let $X\to S\subseteq \mathbb A^1_{\mathbb F_q}\setminus\{0\}$ be a morphism of varieties over $\mathbb F_q$ with a good lift around $t=0$. Let…

Let $R$ be a henselian discrete valuation ring with fraction field $K$ and residue field $k$ of characteristic $p>0$, and let $S=\operatorname{Spec}(R)$. For a proper morphism…