68 problems

Efrat's arithmetical Elementary Type Conjecture. There is such a decomposition where, for each $1\leq i\leq n$, one of the following holds: $G_i\cong\mathbb{Z}_p$; $G_i$ is the dec…

Analytic version of Jonsson–Mustaţă's conjecture. If $c_o^{\mathfrak{q}}(\varphi)<+\infty$, then

Let $F$ be a strongly NIP ordered field. Such a field is almost real closed if it admits a henselian valuation whose residue field is real closed. Shelah–Hasson conjecture for orde…

Let $X$ be a klt variety and let $\mathfrak{a}_\bullet$ be a graded sequence of ideals on $X$ with finite log canonical threshold $\operatorname{lct}(\mathfrak{a}_\bullet)$. A valu…

Let $\mathcal{O}_K$ be a discrete valuation domain with field of fractions $K$ and residue field $k$, and let $v$ be the normalized valuation on $K$. Let $E_1/K$ and $E_2/K$ be ell…

Optimal Destabilization Conjecture. If $\delta(X)\leq 1$, then there exists a divisor $E$ over $X$ computing $\delta(X)$, that is,

McMullen's density conjecture. The linear span

Saturated-valuation conjecture. Under these assumptions, such a strongly positive unipotent bicrystal exists and the valuation-induced map identifies the normal crystal of…

Minimal-valuation local semistable reduction conjecture. $\mathcal{E}$ admits local semistable reduction at any minimal valuation $v$ on $k(X)$ centered on $\overline{X}$.

Let $X$ be a manifold, and for each open subset $U\subset X$ let ${\cal S}_i(U)$ be the space of continuous valuations on $U$. These spaces form a presheaf ${\cal S}_{i,X}$ on $X$.…

Let $V$ have dimension $n$, let $Val_i^\infty(V)$ denote the degree-$i$ smooth translation-invariant valuations, and identify valuations represented by functions on the relevant Gr…

Mixed hard Lefschetz and Hodge–Riemann conjecture. $Val^\infty(V)$ satisfies the mixed hard Lefschetz theorem for the subset ${\cal K}$, and it satisfies the mixed Hodge–Riemann re…

Roé-Urbinati's conjecture. The valuation $v$ is b-divisorial if and only if

Let $\mathcal{O}_2=k[x_1,x_2]$ and let $\mathcal{I}\triangleleft\mathcal{O}_2$ be a finite-codimensional, integrally closed ideal. Let $\mathcal{O}$ denote the ambient module in th…

Fix a model $X_\pi$ of $X$, a point $p\in X_\pi$ whose local ring $\mathcal{O}_{X_\pi,p}$ is regular of dimension $c\le \dim X$, and a regular system of parameters at $p$. Let…

Let $X$ be a normal variety, let $\xi$ be a seminorm on its function field, and let $D_\xi$ be the associated b-divisor. Recovery from a nonzero b-divisor. Whenever $D_\xi\ne 0$, t…

Let $X$ be a normal variety and let $cti$ be a seminorm on its function field, with associated b-divisor $D_\xi$. A valuation is b-divisorial when its associated b-divisor recovers…

Let $X\xrightarrow{\pi}Y\ni p$ be a Fano fibration over an affine pointed variety. Let $v$ be a quasi-monomial valuation of $K(X)$ whose center is supported inside $\pi^{-1}(p)$, a…

Jonsson–Mustaţă's weighted conjecture. The weak version asserts that there exists a quasi-monomial valuation $v\in\mathrm{Val}_X^*$ computing…

Order-or-valuation conjecture. Every unstable NIP field admits either a definable field order or a definable non-trivial valuation.

Let $K$ be a number field, let $p$ be a fixed prime, and let $L=K(\theta)$ be generated by an arbitrary element $\theta$. A dissection is one of the procedures used in Ore's method…

McMullen's conjecture. Every monotone translation invariant valuation that is homogeneous of any order $p$ is given by a mixed volume.

Minimality conjecture. If $\mathrm{F}(p)$ is trivial, then the integer $k$ in equation (mult) is minimal such that

Minimum-bubble conjecture. There are a simple normal-crossings model $\varphi\colon\widetilde{X}\to X$ and a real number $a_{\rm min}$ with $0<a_{\rm min}<a_1(\varphi)$ such that,…

Let $A$ be a domain equipped with an injective well-ordered valuation $\nu$, and let $gr A$ denote the associated graded algebra of the filtration on $A$ induced by $\nu$. A family…