Volumes of Line Bundles as Periods

Definitions and mathematical statement

Let $k$ denote the intended ground field, which the supplied statement does not specify. For a smooth projective variety $X$ over $k$, of dimension $n$, and a line bundle $L$ on $X$, define \[ \operatorname{vol}_X(L)=\limsup_{m\to\infty}\frac{n!}{m^n}\dim_k H^0\!\left(X,L^{\otimes m}\right), \] where $m$ ranges over positive integers. No assumption that $L$ is ample, effective, or big is included in the source question.

Under the usual real-period convention, a real number $p$ is a period if there exist a positive integer $r$, a semialgebraic set $D\subseteq{\mathbb{R}}^r$ defined over ${\mathbb{Q}}$, and a rational function $f\in{\mathbb{Q}}(t_1,\ldots,t_r)$ such that \[ p=\int_D f(t_1,\ldots,t_r)\,{\,\mathrm{d}} t_1\cdots{\,\mathrm{d}} t_r, \qquad \int_D |f(t_1,\ldots,t_r)|\,{\,\mathrm{d}} t_1\cdots{\,\mathrm{d}} t_r<\infty. \] Here semialgebraic means describable by a finite Boolean combination of polynomial equalities and inequalities; being defined over ${\mathbb{Q}}$ means these polynomials have rational coefficients. The integral is taken where $f$ is defined, with any undefined subset of $D$ required to have measure zero. Write $\mathcal P_{{\mathbb{R}}}$ for this set of real periods.

With this convention, the supplied question is whether \[ \operatorname{vol}_X(L)\in\mathcal P_{{\mathbb{R}}} \] for every smooth projective variety $X$ over the intended ground field and every line bundle $L$ on $X$. The source does not specify the ground field, its characteristic, or a definition of period. Thus this is an explicit conditional formulation under the real-period convention above; the intended ground-field scope and period convention require confirmation from the source.

Short English statement

Is the asymptotic growth coefficient of the spaces of global sections of every line bundle on every smooth projective variety a period? The source supplies neither a ground field nor a definition of period; the formulation above uses the usual real-period convention and leaves the ground-field scope unresolved.

Sources

• Supplied local registry record: data/unsolvedmath/problems.json, permanent UnsolvedMath ID $30001602$, title "Volumes of Line Bundles as Periods," source problem code catalogue entry. The supplied record labels the statement exact and reports a manual extraction review; these are supplied provenance claims, not independent verification here.
• Supplied source citation: Mini-Workshop: Linear Series on Algebraic Varieties (2011), Oberwolfach Reports, pp. 2613--2650, DOI 10.4171/owr/2010/45. https://doi.org/10.4171/owr/2010/45. The record attributes the question to workshop open-problem discussions, lists proposed year $2010$, and names no individual proposer.
• Supplied literature reference: https://doi.org/10.1007/s00208-012-0859-0.
• Supplied literature reference: https://www.sciencedirect.com/science/article/pii/S0022404926001684.
• Supplied literature assessment dated exactly 2026-08-21: the record labels the problem open and states that the period-realization question remains explicitly open, with recent examples providing evidence and potential tests rather than a general resolution. This is supplied dated curation derived from the OWR collection, not an independently verified assessment. Its time-sensitive status and cited claims require independent verification.