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On the Bezrukavnikov-Kaledin quantization of symplectic varieties in characteristic $p$

Ekaterina Bogdanova, Vadim Vologodsky

TL;DR

We address Frobenius-constant quantizations of smooth symplectic varieties in characteristic $p>2$ and establish a Morita equivalence after inverting $h$ between Bezrukavnikov–Kaledin quantizations and central reductions of differential-operator algebras. The main engine is a formal-geometry reduction to a group-theoretic problem for the restricted Weyl algebra $A_h$, the construction of a central extension via loop groups, and a key lifting (Basic Lemma) that yields an Azumaya algebra ${ obreak[0]{ abla}}^ lat_h$ on $X'[[h]]$, tying the quantization class to Brauer-group data. When certain cohomology vanishing assumptions hold, ${ obreak[0]{ abla}}^ lat_h$ splits, giving a Morita equivalence between ${ obreak[0]{ abla}}_{X,[ ext{η}],h}(h^{-1})$-modules and ${ obreak[0]{ abla}}_h(h^{-1})$-modules, and the class is computed by the map $ ho$ with $ ext{Br}$-data encoded by $ ext{δ}( ext{γ})$. The paper also develops a framework for ${oldsymbol{G}_m}$-equivariant quantizations and proves a Kubrak–Travkin-type Brauer-class formula in the contracting-action setting, with broader implications for functorial quantization of QCoh$_h$ in a sequel. Collectively, the results refine the BK program in positive characteristic, connect quantizations to central-extensions and Azumaya structures, and pave the way for category-level quantization of coherent sheaves.

Abstract

We prove that after inverting the Planck constant $h$ the Bezrukavnikov-Kaledin quantization $(X, \mathcal{O}_h)$ of symplectic variety $X$ in characteristic $p$ is Morita equivalent to a certain central reduction of the algebra of differential operators on $X$.

On the Bezrukavnikov-Kaledin quantization of symplectic varieties in characteristic $p$

TL;DR

We address Frobenius-constant quantizations of smooth symplectic varieties in characteristic and establish a Morita equivalence after inverting between Bezrukavnikov–Kaledin quantizations and central reductions of differential-operator algebras. The main engine is a formal-geometry reduction to a group-theoretic problem for the restricted Weyl algebra , the construction of a central extension via loop groups, and a key lifting (Basic Lemma) that yields an Azumaya algebra on , tying the quantization class to Brauer-group data. When certain cohomology vanishing assumptions hold, splits, giving a Morita equivalence between -modules and -modules, and the class is computed by the map with -data encoded by . The paper also develops a framework for -equivariant quantizations and proves a Kubrak–Travkin-type Brauer-class formula in the contracting-action setting, with broader implications for functorial quantization of QCoh in a sequel. Collectively, the results refine the BK program in positive characteristic, connect quantizations to central-extensions and Azumaya structures, and pave the way for category-level quantization of coherent sheaves.

Abstract

We prove that after inverting the Planck constant the Bezrukavnikov-Kaledin quantization of symplectic variety in characteristic is Morita equivalent to a certain central reduction of the algebra of differential operators on .

Paper Structure

This paper contains 37 sections, 27 theorems, 227 equations.

Table of Contents

  1. Introduction
  2. Frobenius-constant quantizations.
  3. Differential operators as a Frobenius-constant quantization.
  4. Restricted Poisson structures.
  5. Classification of Frobenius-constant quantizations.
  6. A central reduction of the algebra of differential operators.
  7. Main result.
  8. ${\mathbb G}_m$-equivariant quantizations.
  9. Plan of the proof.
  10. Plan of the paper.
  11. Acknowledgements.
  12. Review of the Bezrukavnikov-Kaledin construction
  13. Darboux Lemma in characteristic $p$.
  14. Quantum Darboux Lemma.
  15. Formal Geometry.
  16. ...and 22 more sections

Key Result

Theorem 1

Let $(X, \omega)$ be a smooth symplectic variety of dimension $2n$ over an algebraically closed field $k$ of characteristic $p>2$, and let $(X, {\mathcal{O}}_h, s)$ be a Frobenius-constant quantization of $(X, \omega)$. Denote by $[\eta] \in H^0_{Zar}(X, \mathop{\mathrm{coker}}\nolimits({\mathcal{O}

Theorems & Definitions (58)

  • Remark 1.1
  • Theorem 1
  • Lemma 3.1
  • proof
  • Remark 3.2
  • proof
  • Proposition 3.3
  • Remark 3.4
  • Corollary 3.5
  • proof
  • ...and 48 more