Lefschetz Properties of Tropical Varieties

Definitions and mathematical statement

The supplied question is: "Can one find other combinatorially interesting families of tropical varieties with the Lefschetz property?"

A precise formulation requires a specified class $\mathscr T$ of tropical varieties, a specified Lefschetz predicate $L$ on $\mathscr T$, and criteria for a family to be combinatorially interesting and additional to previously considered families. With these choices supplied, the research target has the schematic form \[ \text{Exhibit a family }\mathcal F\subseteq\mathscr T \quad\text{such that}\quad \forall X\in\mathcal F,\;L(X), \] together with the required combinatorial interest and novelty conditions. Here $\mathscr T$, $L$, and those conditions are placeholders for missing specifications, not definitions supplied by the source.

The record does not specify the category of tropical varieties, dimensional or compactness hypotheses, the relevant cohomology or algebra, coefficient system, Lefschetz operator, or whether a weak, hard, or other Lefschetz property is intended. It also does not define "combinatorially interesting" or identify the comparison families meant by "other." Consequently, this is an open-ended research question; the supplied evidence does not determine a unique, fully quantified mathematical proposition.

Short English statement

Find additional families of tropical varieties exhibiting a suitable Lefschetz property and meaningful combinatorial structure. The intended Lefschetz property, admissible varieties, and criteria for interest and novelty require clarification. The supplied partially solved label is dated curation, not an independently verified conclusion.

Sources

• Supplied local UnsolvedMath registry record, permanent ID $30004531$, title "Lefschetz Properties of Tropical Varieties," exact source problem code catalogue entry. The record marks the statement as exact and lists proposed year $2020$; no individual proposer is supplied.
• Supplied report citation: Lefschetz Properties in Algebra, Geometry and Combinatorics (2021), Oberwolfach Reports, pp. 1539--1600, DOI 10.4171/owr/2020/31. https://doi.org/10.4171/owr/2020/31. The background describes this report as documenting workshop discussions and open problems.
• Supplied literature reference: Adiprasito--Bjoerner, https://arxiv.org/abs/1401.7301.
• Supplied literature reference: Amini--Piquerez, https://arxiv.org/abs/2007.07826.
• Literature assessment dated exactly 2026-08-22: the supplied curation labels the problem partially solved and reports that tropical Lefschetz theorems are known for smooth tropical varieties and matroidal or Bergman-fan constructions, while the search for further combinatorially interesting families has not been exhausted or classified. This is supplied dated literature triage derived from the OWR curation, not an independently verified assessment. Its time-sensitive status and cited mathematical claims require independent verification.