Lagrangian Varieties from Weyl-Algebra Annihilators

Definitions and mathematical statement

The source asks two questions about the Weyl-algebra reduction construction: is the subvariety of $\operatorname{Spec}Z(W_n)$ defined by its central annihilator Lagrangian, and can every Lagrangian variety arise in this way?

For orientation, the usual Weyl algebra $W_n(k)$ has generators $x_1,\ldots,x_n,\partial_1,\ldots,\partial_n$ with \[ [x_i,x_j]=[\partial_i,\partial_j]=0,\qquad [\partial_i,x_j]=\delta_{ij}. \] In characteristic $p>0$ its center in the standard construction is \[ Z(W_n(k))=k[x_1^p,\ldots,x_n^p,\partial_1^p,\ldots,\partial_n^p], \] so the central spectrum has dimension $2n$. This explains the expected positive-characteristic setting, but the source does not specify the reduction ring, primes, or the object reduced.

If $M$ is the module produced by that construction, its central annihilator would be \[ I_M=\operatorname{Ann}_{Z(W_n)}M, \qquad Y_M=V(I_M)\subseteq\operatorname{Spec}Z(W_n). \] The question asks whether every $Y_M$ from the intended admissible inputs is Lagrangian, and conversely whether each allowed Lagrangian subvariety is some such $Y_M$. The source does not state what those inputs are, including any holonomicity, finite-generation, or characteristic-zero origin hypotheses. This is not asserted for arbitrary Weyl-algebra modules.

For a reduced subvariety of a smooth $2n$-dimensional symplectic variety, the standard Lagrangian condition is pure dimension $n$ and vanishing of the symplectic form on the smooth locus of each irreducible component. The specific symplectic or Poisson structure and treatment of nonreduced central-annihilator schemes must be recovered from the source. The converse also needs a field, class of Lagrangians, meaning of realization, and whether realization is at one prime or coherently after reduction at many primes. No liftability or realization theorem is invented.

Short English statement

The questions ask whether central supports produced by a particular reduced Weyl-algebra construction are Lagrangian and whether they exhaust the intended Lagrangian varieties. The construction's admissible modules, reduction data, symplectic structure, and realization scope are missing.

Sources

• Local UnsolvedMath registry, permanent ID 30000656, code catalogue entry, "Lagrangian Varieties from Weyl-Algebra Annihilators." Proposed year 2007; proposer unspecified.
• Affine Algebraic Geometry (2007), Oberwolfach Reports, pp. 5--82, DOI 10.4171/owr/2007/01; https://doi.org/10.4171/owr/2007/01. This is the sole supplied literature reference.
• Statement status corrected_verified; the supplied note says the original identifies the Weyl-algebra construction and central annihilator and only spectrum/center notation needed restoration. The construction details themselves are not included in this record.
• Literature triage checked 2026-08-21: open. It generically calls the item an unresolved problem/conjecture/direction and reports no authoritative complete resolution in the supplied targeted check. Exact hypotheses and current status require independent verification. Created 2026-08-25; updated 2026-09-16.
• Registry curation by Ulam AI / UnsolvedMath Contributors, CC BY 4.0; the underlying report retains its supplied provenance. Unreviewed statement-only draft.