45 problems

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…

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…

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

Algebraic version of Jonsson–Mustaţă's conjecture. There exists a quasi-monomial valuation $v\in\mathrm{Val}_X^*$ that computes…

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

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…

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…

Jonsson–Mustață's equivalent conjecture. There is a quasimonomial valuation $\nu$ on $\mathcal{O}_o$ that computes $\mathrm{lct}^{\mathfrak{q}}(\mathfrak{b}_{\bullet})$, namely

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 $X$ be a normal affine variety, let $v$ be a fixed polystable negative valuation on $X$, and let $\mathcal M_v$ be the space of compatible Calabi–Yau metrics $\omega$ on $X$ sa…

Let $P\in\mathcal P(\mathbb{Z}^d)$ have dimension $r$, and let the $k$-th discrete intrinsic volume be represented in the binomial basis by … A valuation is combinatorially positiv…

Let $\varphi$ and $\psi$ be plurisubharmonic germs with isolated singularities at $0\in\mathbb{C}^n$. For every divisorial valuation $v$ centered at $0$, write $v(\varphi)$ for its…

Optimal destabilization conjecture. If $(X,D)$ is a K-unstable log Fano pair, then there exists a divisor $E$ over $X$ computing the infimum:

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…

Kotrbatý's Hodge–Riemann conjecture. Let $i\leq n/2$. If $\phi\in Val_i^\infty(W)$ is a non-zero primitive even valuation, then $Q(\phi)>0$; if it is a non-zero primitive odd valua…

Let $v_*$ be a minimizer of the valuation functional $\tilde{\beta}$ for a ${\mathbb{Q}}$-Fano variety $X$, and let $\mathcal{F}_{v_*}$ be the filtration induced by $v_*$. Minimizi…

Characterization conjecture. Let $(\mathbb{Z},+,0,1,v)$ be distal. Then the following are equivalent:

Valuative invariance conjecture. If $\varphi$ and $\psi$ are valuatively equivalent, then their $n$-th Lelong numbers at $0$ are equal: