The generalized Dixmier conjecture and the two-dimensional Jacobian conjecture

Abstract

We prove that the two-dimensional Jacobian conjecture JC₂ implies the generalized Dixmier conjecture DC'₁ for the first Weyl algebra A₁, which states, in one of several equivalent formulations, that every endomorphism of the two-by-two matrix ring Mat₂(A₁) is invertible. For the converse, an endomorphism of 𝔸²_ℂ with Jacobian one gives rise to a family of holonomic A₂-modules over finite fields. One needs to lift this family to an A₂-module in characteristic zero with the prescribed arithmetic support. The existence of such a lift is a special case of Kontsevich's conjecture on holonomic 𝒟-modules with prescribed arithmetic support. We prove that in this special situation such a lift is unique whenever it exists. If these lifts exists for every endomorphism with Jacobian one, then DC'₁ implies JC₂. For every n≥ 3, we construct a Lagrangian subvariety of T^*𝔸ⁿ that provides a counterexample to Kontsevich's conjecture. The counterexamples arise from the graph of a noninvertible symplectic endomorphism of 𝔸⁴ constructed by Long. We also present a different strategy for lifting 𝒟-modules associated with a Jacobian-one endomorphism of 𝔸²_ℂ to characteristic zero via ultraproducts and show that the existence of a characteristic-zero lift in this setting is equivalent to the uniform boundedness of the Berest--Wilson invariants of these A₂-modules, regarded as left A₁-modules. Finally, we study two other conjectures of Kontsevich on arithmetic support, giving a counterexample to one and partial results on the other.

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BibTeX RIS

BibTeX

@misc{kanelbelov2026generalized,
  title = {{The generalized Dixmier conjecture and the two-dimensional Jacobian conjecture}},
  author = {Alexey Kanel-Belov and Timofey Fedorov},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.33919},
  eprint = {2610.33919},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.33919},
}

RIS

TY  - UNPB
TI  - The generalized Dixmier conjecture and the two-dimensional Jacobian conjecture
AU  - Alexey Kanel-Belov
AU  - Timofey Fedorov
PY  - 2026
DA  - 2026/10/24
PB  - arXiv
DO  - 10.48550/arxiv.2610.33919
UR  - https://arxiv.org/abs/2610.33919
AB  - We prove that the two-dimensional Jacobian conjecture JC_2 implies the generalized Dixmier conjecture DC'_1 for the first Weyl algebra A_1, which states, in one of several equivalent formulations, that every endomorphism of the two-by-two matrix ring Mat_2(A_1) is invertible. For the converse, an endomorphism of 𝔸^2_ℂ with Jacobian one gives rise to a family of holonomic A_2-modules over finite fields. One needs to lift this family to an A_2-module in characteristic zero with the prescribed arithmetic support. The existence of such a lift is a special case of Kontsevich's conjecture on holonomic 𝒟-modules with prescribed arithmetic support. We prove that in this special situation such a lift is unique whenever it exists. If these lifts exists for every endomorphism with Jacobian one, then DC'_1 implies JC_2. For every n≥ 3, we construct a Lagrangian subvariety of T^*𝔸^n that provides a counterexample to Kontsevich's conjecture. The counterexamples arise from the graph of a noninvertible symplectic endomorphism of 𝔸^4 constructed by Long. We also present a different strategy for lifting 𝒟-modules associated with a Jacobian-one endomorphism of 𝔸^2_ℂ to characteristic zero via ultraproducts and show that the existence of a characteristic-zero lift in this setting is equivalent to the uniform boundedness of the Berest--Wilson invariants of these A_2-modules, regarded as left A_1-modules. Finally, we study two other conjectures of Kontsevich on arithmetic support, giving a counterexample to one and partial results on the other.
ER  -

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