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D-modules and Solvable Lie Foliations

  • Ameth Ndiaye
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Abstract

Let V be a compact connected manifold and G a simply connected solvable Lie group. We study G-Lie foliations on V from the point of view of π’Ÿ-module theory, following the approach initiated by Dathe. To each singular foliation ℐ, we associate the π’ŸX-module ℳℐ = π’ŸX / π’ŸX Β· ℐ and the derived ring π’Ÿβ„ := Rβ„‹omπ’ŸX(ℳℐ, ℳℐ). We compute the D-irregularity Dβˆ’irr(ℐ) for three classes of solvable Lie foliations: regular homogeneous foliations, the Meigniez foliation with non-polycyclic holonomy group, and an explicit foliation on a compact 5-dimensional manifold. We show that the non-polycyclicity of the holonomy group Ξ“ is reflected in the non-vanishing of higher cohomology groups of π’Ÿβ„, establishing a new connection between the geometry of the holonomy group and π’Ÿ-module invariants.

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Cite

BibTeXRIS
BibTeX
@misc{ndiaye2026dmodules,
  title = {{D-modules and Solvable Lie Foliations}},
  author = {Ameth Ndiaye},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.10583},
  eprint = {2609.10583},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.10583},
}
RIS
TY  - UNPB
TI  - D-modules and Solvable Lie Foliations
AU  - Ameth Ndiaye
PY  - 2026
DA  - 2026/09/06
PB  - arXiv
DO  - 10.48550/arxiv.2609.10583
UR  - https://arxiv.org/abs/2609.10583
AB  - Let V be a compact connected manifold and G a simply connected solvable Lie group. We study G-Lie foliations on V from the point of view of π’Ÿ-module theory, following the approach initiated by Dathe. To each singular foliation ℐ, we associate the π’Ÿ_X-module β„³_ℐ = π’Ÿ_X / π’Ÿ_X Β· ℐ and the derived ring π’Ÿ_ℐ := Rβ„‹om_π’Ÿ_X(β„³_ℐ, β„³_ℐ). We compute the D-irregularity Dβˆ’irr(ℐ) for three classes of solvable Lie foliations: regular homogeneous foliations, the Meigniez foliation with non-polycyclic holonomy group, and an explicit foliation on a compact 5-dimensional manifold. We show that the non-polycyclicity of the holonomy group Ξ“ is reflected in the non-vanishing of higher cohomology groups of π’Ÿ_ℐ, establishing a new connection between the geometry of the holonomy group and π’Ÿ-module invariants.
ER  -