The canonical global quantization of symplectic varieties in characteristic p

2022/11/30 by Ekaterina Bogdanova, Bogdanova, Ekaterina, Dmitry Kubrak +5
Mathematics · #Advanced Algebra and Geometry #Algebraic Geometry (math.AG) #Algebraic Geometry and Number Theory #Category Theory (math.CT) #FOS: Mathematics #Homotopy and Cohomology in Algebraic Topology #Quantum Algebra (math.QA) #Representation Theory (math.RT)

paper · pdf · doi:10.48550/arxiv.2211.17261

openalex publication_date 2022/11/30 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Let X be a smooth symplectic variety over a field k of characteristic p>2 equipped with a restricted structure, which is a class [η] ∈ H⁰(X, Ω¹_X/d\mathcal O_X) whose de Rham differential equals the symplectic form. In this paper we construct a functorial in (X, [η]) formal quantization of the category QCoh(X) of quasi-coherent sheaves on X. We also construct its natural extension to a quasi-coherent sheaf of categories QCoh_h on the product X⁽¹⁾ × \mathbb S of the Frobenius twist of X and the projective line \mathbb S=\mathbb P¹, viewed as the one-point compactification of Spec k[h]. Its global sections over X⁽¹⁾ × \0\ is the category of quasi-coherent sheaves on X. If X is affine, QCoh_h, restricted to X⁽¹⁾× Spf k[[h]], is equivalent to the category of modules over the distinguished "Frobenius-constant" quantization of (X,[η]) defined by Bezrukavnikov and Kaledin.

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