Fourier-Orbit Construction of GKZ-Type Systems for Commutative Linear Algebraic Groups
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866912587146330112 |
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| author | Okuyama, Go |
| author_facet | Okuyama, Go |
| contents | We study GKZ-type D-modules arising from the actions of commutative linear algebraic groups G = TU (where T is a torus and U is unipotent) on a vector space. Building on Hotta's equivariant D-module framework, we formalize a Fourier-orbit construction that recovers the classical toric GKZ system and extends it to mixed torus-unipotent settings. We prove generic holonomicity via a parameter-free symbolic moment ideal and introduce two symbolic tools - the tp-envelope and the symbolic cap - for effective rank analysis and, under mild regularity, exact rank computation. A torus slice yields an explicit lower bound by the normalized lattice volume, explaining sharpness in the pure torus case. Examples exhibit irregular (Airy-type) behavior and resonant non-holonomicity, highlighting new phenomena beyond the toric setting. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2509_03881 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Fourier-Orbit Construction of GKZ-Type Systems for Commutative Linear Algebraic Groups Okuyama, Go Algebraic Geometry 32C38 (Primary), 14M25, 14L30, 33C70 (Secondary) We study GKZ-type D-modules arising from the actions of commutative linear algebraic groups G = TU (where T is a torus and U is unipotent) on a vector space. Building on Hotta's equivariant D-module framework, we formalize a Fourier-orbit construction that recovers the classical toric GKZ system and extends it to mixed torus-unipotent settings. We prove generic holonomicity via a parameter-free symbolic moment ideal and introduce two symbolic tools - the tp-envelope and the symbolic cap - for effective rank analysis and, under mild regularity, exact rank computation. A torus slice yields an explicit lower bound by the normalized lattice volume, explaining sharpness in the pure torus case. Examples exhibit irregular (Airy-type) behavior and resonant non-holonomicity, highlighting new phenomena beyond the toric setting. |
| title | Fourier-Orbit Construction of GKZ-Type Systems for Commutative Linear Algebraic Groups |
| topic | Algebraic Geometry 32C38 (Primary), 14M25, 14L30, 33C70 (Secondary) |
| url | https://arxiv.org/abs/2509.03881 |