Hodge and Order Filtrations on GKZ Modules

Definitions and mathematical statement

Write the intended integer column configuration as $A=(a_1,\ldots,a_m)$, with $a_j\in{\mathbb{Z}}^{d+1}$, solely to explain the semigroup notation in the record. Here $m$ denotes the number of columns; the record does not identify it with its symbol $n$. Define \[ {\mathbb{N}} A=\left\{\sum_{j=1}^{m}u_j a_j : u_j\in{\mathbb{Z}}_{\geq 0}\right\}\subseteq{\mathbb{Z}}^{d+1}. \] For $c=(c_0,\ldots,c_d)\in{\mathbb{N}} A$, the integer $c_0$ is its first component. The notation $M_A^\beta$ denotes the GKZ module with configuration $A$ and parameter $\beta$, as named in the source. Its ambient space, defining convention, and the admissible configurations $A$ are not specified in the supplied record.

An increasing filtration on a module $M$ is a family $(F_pM)_{p\in{\mathbb{Z}}}$ with $F_pM\subseteq F_{p+1}M$. The source uses $F^H$ for a Hodge filtration and $F^{\mathrm{ord}}$ for an order filtration. In a specified cyclic differential-operator presentation $M=D/I$, the usual order filtration is the image of operators of order at most $p$ in $M$, with zero terms for $p<0$. The record supplies neither such a presentation and its normalization nor a Hodge-module realization determining $F^H$.

Recorded assertion, subject to unresolved hypotheses. For the intended configurations and parameters $c\in{\mathbb{N}} A$, the source asserts, for every $p\in{\mathbb{Z}}$, \[ F^H_p M_A^0=F^{\mathrm{ord}}_{p+d}M_A^0, \qquad F^H_p M_A^{-c}=F^{\mathrm{ord}}_{p-n-c_0}M_A^{-c}. \] These are equalities of filtration terms on the respective modules. The symbol $n$, the class of admissible configurations, and any additional restrictions on $c$ remain unspecified. In particular, rank, homogeneity, normality or saturation hypotheses cannot be inferred from the displayed assertion. Consequently, this is a faithful transcription of the proposed identities, not a fully specified universally quantified conjecture.

The supplied background instead discusses a GKZ weight-filtration conjecture and a full saturated-semigroup assertion. It gives no formulation of that assertion or explanation connecting it to these Hodge-filtration identities. That background qualification is retained as unresolved.

Short English statement

The recorded claim says that the Hodge filtrations on $M_A^0$ and $M_A^{-c}$ agree with their order filtrations after shifts by $d$ and $-n-c_0$, respectively, where $c\in{\mathbb{N}} A$. The allowed configurations, the meaning of $n$, and the Hodge structures are unspecified. The supplied partial-solution assessment concerns weight filtrations, whose connection to these identities requires review.

Sources

• Supplied local UnsolvedMath registry record, permanent ID $30002871$, source problem code catalogue entry, title Hodge and Order Filtrations on GKZ Modules; project registry data/unsolvedmath/problems.json. This attribution uses the supplied record; the local registry file was not independently inspected.
• Supplied source citation: Mirror Symmetry, Hodge Theory and Differential Equations (2016), Oberwolfach Reports, DOI 10.4171/owr/2015/22, pp. 1201--1254. https://doi.org/10.4171/owr/2015/22. The literature-source list labels this reference "Oberwolfach Report 22." The record assigns proposed year 2015 and supplies no named proposer.
• Supplied literature reference: Reichelt--Walther, Weight Filtrations on GKZ-Systems. https://arxiv.org/abs/1809.04247.
• Literature assessment dated exactly 2026-08-21: supplied dated curation, not independently verified. It labels the problem partially solved and states that the GKZ weight-filtration conjecture is established in normal and several low-dimensional or simplicial settings, with explicit image and filtration results, while the full saturated-semigroup assertion in the record is not proved generally. The supplied background describes this as literature triage derived from OWR curation and requests independent verification of time-sensitive status and cited claims. Its relationship to the displayed Hodge-filtration identities is unresolved.
• The registry labels the statement corrected_verified and reports manual semantic, typographic, and LaTeX review of a meaning-preserving cleanup. These are supplied editorial labels and do not establish the mathematical identities or resolve their missing hypotheses.