The generalized Dixmier conjecture and the two-dimensional Jacobian conjecture

Source abstract

We prove that the two-dimensional Jacobian conjecture $\mathrm{JC}_2$ implies the generalized Dixmier conjecture $\mathrm{DC}'_1$ 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 $\operatorname{Mat}_2(A_1)$ is invertible. For the converse, an endomorphism of $\mathbb A^2_{\mathbb C}$ 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 $\mathcal D$-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 $\mathrm{DC}'_1$ implies $\mathrm{JC}_2$. For every $n\ge 3$, we construct a Lagrangian subvariety of $T^*\mathbb{A}^n$ that provides a counterexample to Kontsevich's conjecture. The counterexamples arise from the graph of a noninvertible symplectic endomorphism of $\mathbb A^4$ constructed by Long. We also present a different strategy for lifting $\mathcal D$-modules associated with a Jacobian-one endomorphism of $\mathbb A^2_{\mathbb C}$ 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.