Quasimonomial Valuations Computing Asymptotic Log Canonical Thresholds

Definitions and mathematical statement

The supplied question does not specify the hypotheses on $X$, the ground field, or the allowed ideals. Consequently, its precise scope remains unresolved. The following explains the usual valuation-theoretic interpretation conditionally, in a setting where function fields, birational models, log discrepancies, and the valuation formula for log canonical thresholds are defined.

A graded sequence of ideals is a family $\mathfrak a_\bullet=(\mathfrak a_m)_{m\geq 1}$ satisfying \[ \mathfrak a_m\mathfrak a_n\subseteq\mathfrak a_{m+n}\qquad(m,n\geq 1). \] For an integral $X$, a real valuation centered on $X$ is a valuation $\nu:K(X)^\times\to{\mathbb{R}}$, trivial on the ground field when a ground field is specified, whose valuation ring dominates the local ring of its center on $X$. For a nonzero finitely generated ideal $\mathfrak b$ at that center, let $\nu(\mathfrak b)$ be the minimum of the values of a set of local generators. For a sequence for which these values are defined, put \[ \nu(\mathfrak a_\bullet)=\inf_{m\geq 1}\frac{\nu(\mathfrak a_m)}{m}. \] A valuation is quasi-monomial if, on a suitable regular birational model of $X$, it is monomial in regular parameters cutting out the components of a simple normal crossings divisor at a stratum, with positive real weights on the selected parameters. Write $A_X(\nu)$ for its log discrepancy over $X$.

In a setting where the usual valuation formula applies, the weighted asymptotic log canonical threshold is expressed as \[ \operatorname{lct}_{\mathfrak q}(\mathfrak a_\bullet) =\inf_{\nu}\frac{A_X(\nu)+\nu(\mathfrak q)}{\nu(\mathfrak a_\bullet)}, \] with the valuation class and conventions for zero or infinite values determined by that setting. For a valuation with $0<\nu(\mathfrak a_\bullet)<\infty$ and finite numerator, saying that it computes this threshold means \[ \operatorname{lct}_{\mathfrak q}(\mathfrak a_\bullet) =\frac{A_X(\nu)+\nu(\mathfrak q)}{\nu(\mathfrak a_\bullet)}. \]

The source question is: for every graded sequence $\mathfrak a_\bullet$ on $X$ and every fixed ideal $\mathfrak q$ in the intended, but unspecified, class, does there exist a quasi-monomial valuation $\nu$ computing $\operatorname{lct}_{\mathfrak q}(\mathfrak a_\bullet)$? The missing ambient hypotheses, admissibility conditions, and boundary conventions must be recovered before this becomes a fully specified universal statement. The question asks for existence of such a valuation; it does not state that every minimizing valuation is quasi-monomial.

Short English statement

The question asks whether the weighted asymptotic log canonical threshold of every admissible graded sequence of ideals is attained by a quasi-monomial valuation. The supplied record leaves the geometric setting and degenerate cases unspecified. Its supplied dated assessment reports partial results, including an unweighted case, and describes a general strong statement for arbitrary $\mathfrak q$ as open; neither that assessment nor its relationship to the displayed existence question is independently verified here.

Sources

• Local UnsolvedMath registry, permanent ID $30001573$, source problem code catalogue entry, title "Quasimonomial Valuations Computing Asymptotic Log Canonical Thresholds." The supplied record belongs to the Oberwolfach Reports Open Problems collection and gives proposed year 2010; no individual proposer is supplied.
• Supplied original report citation: Komplexe Analysis (2011), Oberwolfach Reports, DOI 10.4171/owr/2010/39, pp. 2283--2333. https://doi.org/10.4171/owr/2010/39. This is also the supplied literature-source link labeled "original OWR report." The report title's year differs from the year in the DOI and proposed-year metadata; the supplied citation is retained without resolution.
• Supplied literature source: Xu, "A minimizing valuation is quasi-monomial." https://annals.math.princeton.edu/2020/191-3/p06.
• Supplied literature source: Jonsson--Mustata, "asymptotic invariants." https://www.numdam.org/item/10.5802/aif.2746.pdf. The author string has corrupted diacritic markup in the record and is rendered here without accents.
• Literature assessment dated exactly 2026-08-21: the supplied curation labels the problem partially_solved and reports quasi-monomial minimizing valuations in important cases, including the unweighted case, while describing the general strong statement for arbitrary $\mathfrak q$ as open. This is supplied dated curation, not an independently verified literature assessment. The background expressly requests independent verification of time-sensitive status and cited claims.
• The record labels the statement corrected_verified and reports that its mathematical content was checked against the official Oberwolfach report, with notation, context, and question form normalized. This is a supplied verification claim; no source visit or independent verification is asserted here.