Convex Geometry and Limits of Asymptotic Higher Cohomology

Definitions and mathematical statement

Write $n=\dim X$, and let $D$ denote the divisor intended in the source. Let $k$ denote the ground field, which the supplied record does not specify. Whenever the sheaves $\mathcal O_X(mD)$ are defined and their cohomology groups are finite-dimensional over $k$, set \[ \widehat h^i(D):=n!\limsup_{\substack{m\to\infty\\m\in{\mathbb{Z}}_{>0}}}\frac{\dim_k H^i(X,\mathcal O_X(mD))}{m^n}. \] Here $i$ is a cohomological degree; the question concerns higher cohomology, hence positive degrees, although the source does not explicitly specify the range of $i$.

The source asks two related questions:

1. Is there a natural convex-geometric construction, analogous to the Newton--Okounkov interpretation of divisor volume, which realizes $\widehat h^i(D)$ as a Euclidean volume?
2. For every admissible $X,D,i$, does the finite limit \[ \lim_{\substack{m\to\infty\\m\in{\mathbb{Z}}_{>0}}}\frac{\dim_k H^i(X,\mathcal O_X(mD))}{m^n} \] exist, so that \[ \widehat h^i(D)=n!\lim_{\substack{m\to\infty\\m\in{\mathbb{Z}}_{>0}}}\frac{\dim_k H^i(X,\mathcal O_X(mD))}{m^n}? \]

The quantifier "every admissible $X,D,i$" remains conditional: the supplied record does not state the admissible geometric class, ground-field hypotheses, or divisor hypotheses. In particular, projectivity, smoothness, and whether $D$ is Cartier are not specified. If only certain positive multiples of $D$ define the intended sheaves, the indexing sequence must also be specified; the displayed source formula uses all positive integers. The convex-geometric question likewise leaves the object, its dimension, its permitted auxiliary choices, and the normalization of Euclidean volume unspecified. These omissions prevent a fully determined universal formulation.

Short English statement

The problem asks for a natural convex-geometric volume interpretation of asymptotic higher cohomology and asks whether its defining $\limsup$ is always an ordinary limit. The intended geometric hypotheses and the meaning of the proposed volume construction require clarification. The supplied registry labels the problem open; this draft does not independently verify that status.

Sources

• Supplied local UnsolvedMath registry record, permanent ID $30003591$, source problem code catalogue entry, titled "Convex Geometry and Limits of Asymptotic Higher Cohomology." The record gives proposed year 2017 and names no individual proposer.
• Supplied report citation: Mini-Workshop: Positivity in Higher-dimensional Geometry: Higher-codimensional Cycles and Newton--Okounkov Bodies (2018), Oberwolfach Reports, pp. 2631--2657, DOI 10.4171/owr/2017/43. Supplied source link: https://doi.org/10.4171/owr/2017/43. The background describes the report as documenting workshop discussions and open problems.
• Supplied literature reference: "Kuronya, asymptotic cohomological functions," https://arxiv.org/abs/math/0501491.
• Supplied literature reference: "Li, asymptotic cohomological functions and volumes," https://arxiv.org/abs/2607.09021.
• Supplied literature reference: "original OWR report," https://doi.org/10.4171/OWR/2017/43.
• Literature assessment dated exactly 2026-08-21: the supplied curation labels the problem open, attributes continuity and numerical invariance of asymptotic cohomological functions to Kuronya, states that the normalized higher-cohomology $\limsup$ is not known to be a limit in general, and distinguishes the cited 2026 progress on volume limits from this higher-cohomology question. This is supplied dated curation derived from the OWR material, not an independently verified assessment; its time-sensitive status and cited claims require independent checking.