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PreprintCC BY 4.0arXiv

Geometry of holonomic modules over the first Weyl algebra

  • A. V. Petukhov
  • M. D. Sulimov
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Abstract

We give a geometric description of the formal local data of holonomic modules over the first Weyl algebra, the algebra of differential operators in one variable. The exponential part of the associated formal meromorphic connection is encoded by a truncated jet of an affine plane curve at infinity, and the regular part by a regular formal meromorphic connection. We prove that these geometric data uniquely determine the `elementary' blocks in the formal decomposition of holonomic modules. The resulting correspondence is equivariant under automorphisms of the first Weyl algebra, including the Fourier transform.

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Cite

BibTeXRIS
BibTeX
@misc{petukhov2026geometry,
  title = {{Geometry of holonomic modules over the first Weyl algebra}},
  author = {A. V. Petukhov and M. D. Sulimov},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.35447},
  eprint = {2610.35447},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.35447},
  note = {CC BY 4.0},
}
RIS
TY  - UNPB
TI  - Geometry of holonomic modules over the first Weyl algebra
AU  - A. V. Petukhov
AU  - M. D. Sulimov
PY  - 2026
DA  - 2026/10/26
PB  - arXiv
DO  - 10.48550/arxiv.2610.35447
UR  - https://arxiv.org/abs/2610.35447
AB  - We give a geometric description of the formal local data of holonomic modules over the first Weyl algebra, the algebra of differential operators in one variable. The exponential part of the associated formal meromorphic connection is encoded by a truncated jet of an affine plane curve at infinity, and the regular part by a regular formal meromorphic connection. We prove that these geometric data uniquely determine the `elementary' blocks in the formal decomposition of holonomic modules. The resulting correspondence is equivariant under automorphisms of the first Weyl algebra, including the Fourier transform.
ER  -