Algebraic Realizability of Tropical Varieties

Definitions and mathematical statement

For a variety $V\subset(K^\times)^n$ over a field with a real-valued valuation $\operatorname{val}$, one standard tropicalization is \[ \operatorname{Trop}(V)=\overline{\{(\operatorname{val}x_1,\ldots,\operatorname{val}x_n):x\in V\}}, \] with the appropriate valued extension if required. Tropical cycles are commonly represented by weighted rational polyhedral complexes satisfying balancing at codimension-one faces; weights encode more data than the underlying set.

For a complex variety $V\subset({\mathbb{C}}^\times)^n$, its amoeba is \[ \mathcal A(V)=\{(\log|z_1|,\ldots,\log|z_n|):z\in V\}\subset{\mathbb{R}}^n. \] An amoeba-limit correspondence requires a family of varieties, a rescaling and a specified topology, with a separate convention for recovering multiplicities.

The source asks for appropriate tropical counterparts of algebraic varieties and a precise correspondence with tropical cycles/varieties or limits of amoebas. This includes specifying a forward tropicalization construction, determining which tropical objects are algebraically realizable, and identifying the appropriate limiting correspondence. A realization assertion would take the form $\exists V:\operatorname{Trop}(V)=T$ for a specified tropical object $T$ in a specified field/embedding class; the record does not assert that every balanced complex is realizable or that the correspondence is one-to-one.

No base field, valuation, ambient torus or compactification, admissible variety/cycle class, multiplicity convention, family, rescaling or limiting topology is supplied. The displayed standard constructions identify the mathematical setting without choosing a universal realization theorem. The unrestricted request is retained as a research program, alongside the supplied qualification that systematic constructions and correspondence theorems already exist in broad classes.

Short English statement

Specify the correspondence between algebraic varieties, tropical cycles and rescaled amoeba limits, including the algebraic realization problem. The source gives no precise class, field or limit convention, and does not assert a bijection or universal realizability.

Sources

• Supplied report: Classical Algebraic Geometry (2009), Oberwolfach Reports, DOI 10.4171/owr/2008/26, pp. 1425--1474; proposed year 2008. https://doi.org/10.4171/owr/2008/26.
• Supplied reference: Mikhalkin, Tropical geometry and its applications, https://arxiv.org/abs/math/0408099.
• Local UnsolvedMath registry, permanent ID 30000976, catalogue entry; created 25 August 2026, updated 16 September 2026. Its exact-statement note reports manual confirmation of coherence and absence of layout artifacts. The supplied 21 August 2026 triage labels the record partially solved, reports systematic tropicalizations and broad correspondence results, and characterizes the all-counterparts question as a continuing program. This dated assessment is not independently updated here.