Deformations of Non-Gorenstein Toric Fano Varieties

Definitions and mathematical statement

The source poses a research programme rather than a conjecture with a specified conclusion. Its ground field, characteristic, dimension, deformation bases, and precise convention for $\mathbb Q$-Gorenstein families are not specified. The definitions below give a conventional framework for interpreting the question; these choices require confirmation against the source.

Let $k$ denote the unspecified ground field. A toric variety $X$ over $k$ is a normal variety containing an algebraic torus as a dense open subset, with the torus action extending to $X$. Under the convention used here, a toric Fano variety is projective, its canonical divisor $K_X$ is $\mathbb Q$-Cartier, and $-K_X$ is ample. A divisor is $\mathbb Q$-Cartier if some positive integral multiple is Cartier. The phrase "with non-Gorenstein singularities" is interpreted as requiring at least one point $x\in X$ for which the local ring $\mathcal O_{X,x}$ is not Gorenstein. Whether the source intends any narrower singularity class is unspecified.

A deformation of $X$ over a pointed $k$-scheme $(S,s)$, with residue field $k$ at $s$, consists of a flat proper morphism \[ \pi\colon\mathcal X\longrightarrow S, \qquad \mathcal X_s\cong X. \] For a coherent sheaf $\mathcal F$, write $\mathcal F^{**}=\mathcal H om(\mathcal H om(\mathcal F,\mathcal O),\mathcal O)$. Whenever a suitable relative dualizing sheaf exists, write \[ \omega_{\mathcal X/S}^{[m]}= \left(\omega_{\mathcal X/S}^{\otimes m}\right)^{**} \qquad(m\geq 1). \] One common strong convention for a $\mathbb Q$-Gorenstein deformation requires these reflexive powers to be flat over $S$ and compatible with base change, and requires $\omega_{\mathcal X/S}^{[r]}$ to be invertible for some positive integer $r$, locally on the family. On normal spaces where relative canonical divisors are defined, the latter condition expresses that $K_{\mathcal X/S}$ is $\mathbb Q$-Cartier. The source does not specify the hypotheses on families needed for this formulation or whether it intends this base-change convention or a weaker relative $\mathbb Q$-Cartier condition.

Problem. Develop a combinatorial and deformation-theoretic description of the $\mathbb Q$-Gorenstein deformations of toric Fano varieties having non-Gorenstein singularities. In a setting where toric varieties are described by fans in lattices, the requested combinatorial description should relate the toric data of $X$ to its $\mathbb Q$-Gorenstein deformation theory. The original question is: "What can be said about Q-Gorenstein deformations of toric Fano varieties whose singularities are not Gorenstein?" Neither a specific classification target nor a necessary-and-sufficient criterion is supplied. The statement does not impose torus equivariance on the deformations; whether equivariant deformations or general deformations are intended remains unresolved.

Short English statement

Describe, using toric combinatorics and deformation theory, the $\mathbb Q$-Gorenstein deformations of toric Fano varieties with non-Gorenstein singularities. The source leaves the ground field, allowed families, precise $\mathbb Q$-Gorenstein convention, and expected form of the description unspecified. This is a statement-only research-problem draft requiring mathematical review.

Sources

• Local UnsolvedMath registry, supplied record with permanent ID $30004250$, title "Deformations of Non-Gorenstein Toric Fano Varieties," source problem code catalogue entry, in the Oberwolfach Reports Open Problems collection. The record gives proposed year $2019$ and names no proposer.
• Supplied primary report citation: Toric Geometry (2020), Oberwolfach Reports, pp. 2671--2726, DOI 10.4171/owr/2019/43. https://doi.org/10.4171/owr/2019/43. The supplied background describes Oberwolfach Reports as documenting workshops and their open-problem discussions. The citation's $2020$ title date and the record's $2019$ proposed year are retained as supplied.
• Supplied literature source, labelled "Toric deformation spaces": https://doi.org/10.1090/tran/8636.
• Supplied literature source, labelled "Toric Q-Gorenstein singularities": https://arxiv.org/abs/alg-geom/9403003.
• Literature assessment checked at 2026-08-22: the supplied curation labels the problem partially_solved and reports that $\mathbb Q$-Gorenstein deformation theory and classifications exist in restricted singularity classes, while no general solution for non-Gorenstein toric Fano varieties was located. This is supplied dated curation derived from the OWR record, not an independently verified literature assessment. Its time-sensitive status and cited claims require independent verification.
• The registry labels the statement corrected_verified and reports that mathematical content was preserved while notation, typography, and sentence structure were repaired for publication. These are supplied editorial qualifications, not mathematical verification of this draft.