Hodge Numbers from Landau–Ginzburg Fiber Components

Definitions and mathematical statement

The supplied question concerns a smooth Fano variety $X$ of dimension $n$, equipped with a toric Landau--Ginzburg model $f_X$. Under the usual complex-geometric interpretation, a smooth Fano variety is a smooth projective variety over ${\mathbb{C}}$ whose anticanonical line bundle is ample. For $n\geq 1$, its relevant Hodge number is \[ h^{1,n-1}(X)=\dim_{{\mathbb{C}}}H^{n-1}(X,\Omega_X^1), \] where $\Omega_X^1$ is the sheaf of algebraic differential one-forms. The base field and the dimension range are not explicitly specified in the supplied record; this complex-geometric interpretation requires confirmation.

To express the proposed fiber count, suppose a chosen compactification of $f_X$ is a morphism $\overline f_X\colon Y\to B$. Let $R\subseteq B$ be the set of parameters of the reducible fibers included in the count. If $R$ is finite and $r_t$ denotes the number of irreducible components of the fiber over $t$, counted without multiplicity, the supplied description becomes \[ k_{f_X}=\sum_{t\in R}r_t-\#R =\sum_{t\in R}(r_t-1). \] This formula interprets "components" as irreducible components. The record does not specify the compactification, which fibers are included, whether a fiber at infinity is included, or whether this count is independent of those choices. Finiteness and the component-counting convention also require confirmation.

The proposed mathematical assertion is that, for every smooth Fano variety $X$ of dimension $n$ equipped with a toric Landau--Ginzburg model $f_X$, \[ h^{1,n-1}(X)=k_{f_X}. \] Here the meaning of "toric Landau--Ginzburg model" and the allowed compactification choices must be supplied before this becomes a fully specified universal assertion. In particular, the record does not determine whether the equality is required for every admissible compactification or for a distinguished choice. These unresolved qualifications are part of this statement-only draft.

Short English statement

The question asks whether a particular Hodge number of a smooth Fano variety equals the total number of extra components in the reducible fibers of a compactified toric Landau--Ginzburg model: a fiber with $r$ components contributes $r-1$. The precise model and compactification conventions remain unspecified. The supplied dated curation labels the problem partially solved; that status has not been independently verified.

Sources

• Local UnsolvedMath registry: supplied source record for permanent ID $30003864$, source problem code catalogue entry, in the Oberwolfach Reports Open Problems collection. No individual proposer is identified; the proposed-year field is $2018$.
• Supplied report citation: Subgroups of Cremona Groups (2019), Oberwolfach Reports, pp. 1685--1743, DOI 10.4171/owr/2018/28. https://doi.org/10.4171/owr/2018/28. The supplied background describes Oberwolfach Reports as documenting mathematical workshops and their open-problem discussions. No exact page or passage for this question is supplied.
• Supplied literature source: Cheltsov--Przyjalkowski, "KKP for Fano threefolds." https://doi.org/10.4213/tm4452.
• Supplied literature source: Przyjalkowski, "KKP conjectures in lower dimensions." https://ems.press/content/serial-article-files/46750.
• Supplied literature assessment, checked on 2026-08-21: the status is recorded as partially solved; the Hodge-number mirror relation is said to be proved for several classes, a 2025 paper is said to prove the KKP conjecture for smooth Fano threefolds, and the extracted assertion for all smooth Fano varieties is said to remain open in higher dimensions. This is supplied dated curation derived from the OWR record, not an independently verified literature assessment. The source explicitly requests independent verification of time-sensitive status and cited claims.
• The registry labels the statement corrected_verified and reports manual semantic, typographic, and LaTeX review asserting self-containment and preservation of the mathematical target. This is supplied review metadata, not independent verification of the mathematical formulation or its truth.