Quantization of restricted Lagrangian subvarieties in positive characteristic

TL;DR

This paper studies the conditions under which line bundles on Lagrangian subvarieties in positive characteristic can be quantized, linking geometric properties to cohomological obstructions in the context of restricted Lie algebra structures.

Contribution

It establishes a criterion involving a cohomology class for the existence of quantizations of line bundles on Lagrangian subvarieties within restricted Poisson varieties.

Findings

A cohomology class determines quantizability of line bundles.

Quantization exists if and only if a specific cohomology class vanishes.

The work connects algebraic structures with geometric quantization in positive characteristic.

Abstract

Bezrukavnikov and Kaledin introduced quantizations of symplectic varieties X in positive characteristic which endow the Poisson bracket on X with the structure of a restricted Lie algebra. We consider deformation quantization of line bundles on Lagrangian subvarieties Y of X to modules over such quantizations. If the ideal sheaf of Y is a restricted Lie subalgebra of the structure sheaf of X, we show that there is a certain cohomology class which vanishes if and only if a line bundle on Y admits a quantization.