Maps from Varieties to Okounkov Bodies

Definitions and mathematical statement

Write $\Delta(X,L,\nu)$ for the Okounkov body associated with a variety $X$, a line bundle $L$, and valuation data $\nu$. For orientation, in a section-valuation construction with $n=\dim X$ and valuation vectors in ${\mathbb{Z}}^n$, one uses \[ \Gamma(X,L,\nu)=\{(m,\nu(s)):m\in{\mathbb{Z}}_{>0},\ 0\ne s\in H^0(X,L^{\otimes m})\} \] and \[ \Delta(X,L,\nu)=\overline{\operatorname{conv}\!\left\{\frac{\nu(s)}{m}:(m,\nu(s))\in\Gamma(X,L,\nu)\right\}}\subseteq{\mathbb{R}}^n. \] This presentation is conditional on having data for which that construction is defined; it does not supply hypotheses missing from the source.

The supplied question is: does there exist, in general, a reasonable map \[ f:X\longrightarrow\Delta(X,L,\nu), \] and under what hypotheses on $X,L,\nu$ can such a map be constructed?

The word "reasonable" is not defined in the record. In particular, the source does not specify the field of definition, the class of varieties, the assumptions on $L$ or $\nu$, the underlying points or topology of the domain, or the properties required of $f$. Thus the displayed question is a research formulation whose precise quantifiers and map requirements remain unresolved; mere existence of an arbitrary set map would not specify the intended target.

Short English statement

The problem asks whether a variety admits a suitably meaningful map to its Okounkov body, and which hypotheses permit its construction. The supplied record leaves both "reasonable" and the admissible geometric data unspecified, so it does not determine a single precise existence assertion.

Sources

• Supplied local UnsolvedMath registry record, permanent ID $30002040$, title Maps from Varieties to Okounkov Bodies, source problem code catalogue entry, in the Oberwolfach Reports Open Problems collection. No individual proposer is supplied; the proposed year is recorded as 2012.
• Supplied report citation: Toric Geometry (2013), Oberwolfach Reports, DOI 10.4171/owr/2012/21, pp. 1231--1306. https://doi.org/10.4171/owr/2012/21. The supplied background describes these reports as documenting workshops and their open-problem discussions. The citation's title year is retained alongside the recorded proposed year without resolving their relationship.
• Harada--Kaveh, Integrable systems, toric degenerations and Okounkov bodies. https://arxiv.org/abs/1205.5249.
• Murata--Bossinger, Maps to toric varieties and toric degenerations. https://arxiv.org/abs/2210.13137.
• Literature assessment dated exactly 2026-08-21: the supplied curation labels the problem "partially solved" and states that Harada--Kaveh construct the requested map when a suitable toric degeneration or finitely generated Okounkov semigroup exists, that later work extends the construction, and that no general construction for arbitrary varieties is established. This is supplied dated literature triage derived from the OWR curation, not an independently verified assessment; its status and cited claims require independent verification.
• The supplied registry labels the statement corrected_verified and reports that the official report was checked, that the complete question was confirmed and preserved, and that undelimited notation was repaired. These are supplied source-verification claims, not verification performed for this draft.