Necessary Conditions for Laurent-Polynomial Fano Mirrors

Definitions and mathematical statement

For a field $k$ and a specified number $n$ of variables, a Laurent polynomial is an element \[ f\in k[x_1^{\pm1},\ldots,x_n^{\pm1}],\qquad f=\sum_{m\in A}c_m x_1^{m_1}\cdots x_n^{m_n}, \] where $A\subset{\mathbb{Z}}^n$ is finite and $c_m\in k$. Under the usual algebraic convention, a Fano manifold is a smooth projective geometrically integral variety $X$ whose anticanonical line bundle $\mathcal O_X(-K_X)$ is ample. The source does not specify the base field, the number of variables, the dimension of $X$, or any additional restrictions on $f$ or $X$; these conventions therefore do not resolve the intended geometric setting.

The source question is: What necessary conditions must a Laurent polynomial $f$ satisfy in order to be mirror to a Fano manifold?

Once a geometric setting and a precise mirror relation $\mathcal M(f,X)$ have been specified, the requested conditions can be expressed as properties $P$ of Laurent polynomials for which \[ \forall f\in k[x_1^{\pm1},\ldots,x_n^{\pm1}],\qquad \left(\exists X\text{ a Fano manifold with }\mathcal M(f,X)\right) \Longrightarrow P(f). \] Here $\mathcal M$ is only a placeholder for the unspecified meaning of being mirror; no definition of that relation is supplied or asserted. The class of admissible $X$, including any requirement that $\dim X=n$, must also be specified before this schema becomes a determinate mathematical problem. The question asks for necessary conditions, without asserting their sufficiency or asking explicitly for a complete characterization.

Short English statement

Determine restrictions that every Laurent polynomial mirror to a Fano manifold must satisfy. The supplied question leaves the meaning of mirror and the geometric setting unspecified, so a precise version requires those choices. Its supplied status is partially solved; this is a dated curation label, not an independently verified conclusion.

Sources

• Local UnsolvedMath registry, permanent ID $30001912$, source problem code catalogue entry, title Necessary Conditions for Laurent-Polynomial Fano Mirrors, in the Oberwolfach Reports Open Problems collection. The record gives proposed year 2011 and names no individual proposer.
• Supplied report citation: Complex Algebraic Geometry (2012), Oberwolfach Reports, DOI 10.4171/owr/2011/47, pp. 2669--2733. https://doi.org/10.4171/owr/2011/47. The background describes these reports as documenting mathematical workshops and their open-problem discussions. The supplied citation uses 2012, whereas the proposed year and DOI contain 2011; this discrepancy is retained without resolution.
• Supplied literature reference: Coates--Corti--Galkin--Kasprzyk, quantum periods. https://arxiv.org/abs/1303.3288.
• Supplied literature reference: Coates--Kasprzyk--Pitton--Tveiten, maximally mutable Laurent polynomials. https://arxiv.org/abs/2107.14253.
• Supplied literature reference: original OWR report. https://doi.org/10.4171/owr/2011/47.
• Supplied literature assessment, checked at 2026-08-21: the status is partially solved. The assessment states that period, regularity, and extremality conditions and large families of Laurent-polynomial mirrors are established, but that there is no complete necessary-and-sufficient characterization of mirrors to Fano manifolds. This is supplied dated curation derived from the OWR record, not independently verified here; its time-sensitive status and cited claims require independent checking.
• The registry labels the statement corrected_verified and reports that its mathematical content was checked against the official report, with notation, context, and question form normalized without changing the source-level claim. This is supplied verification metadata, not a claim that the report was consulted for this draft.