30 problems

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…

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

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 $X$ 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…

Let $k$ be a perfect field, let $X$ be a smooth $k$-variety, and let $\mathcal{E}$ be an overconvergent $F^a$-isocrystal on $X$, where $F^a$ denotes a Frobenius structure of order…

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

The semistable-reduction conjecture. There exists a ${\cal G}\otimes_{{\cal O}_F}{\cal O}_E$-equivariant blowing up in the special fiber

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…

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…

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 discrete valuation field with residue field $k$, let $\ell$ be different from the characteristic of $k$, and let $X$ be a proper smooth scheme over $K$. For…

Log semistable reduction over a hollow curve. Possibly after passing to a Kummer étale cover of $X$, there exists a commutative square

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

Local modification conjecture. The local versions of the semistable and log-smooth modification conjectures assert that such a modification exists locally along a semivaluation on…

Semistable modification conjecture. After a ground field extension, any model can be blown up to a semistable one.

Semistable reduction conjecture. After a finite separable extension of the ground field, every smooth proper variety over $k$ possesses a semistable proper model over $R$.

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…

Hard Lefschetz-type conjecture. For every $\ell\neq p$ in the $\mathbb Q_\ell$ case, respectively for all but finitely many $\ell\neq p$ in the $\mathbb F_\ell$ and…

Let $K$ be a complete discrete valuation field of characteristic zero with perfect residue field of characteristic $p$, and let a smooth proper algebraic variety over $K$ be a smoo…

Abramovich–Karu smoothness conjecture. The space $\widetilde{X}$ can be made smooth.

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…