Quantization on Algebraic Curves with Frobenius-Projective Structure

TL;DR

This paper explores the connection between deformation quantizations and Frobenius-projective structures on algebraic curves in positive characteristic, providing a canonical construction of Frobenius-constant quantizations and extending results to higher dimensions.

Contribution

It introduces a canonical method to construct Frobenius-constant quantizations on algebraic curves using Frobenius-projective structures, extending complex case results to positive characteristic.

Findings

Constructs Frobenius-constant quantizations from Frobenius-projective structures.

Establishes a positive characteristic analogue of known complex results.

Provides a higher-dimensional generalization of the main construction.

Abstract

In the present paper, we study the relationship between deformation quantizations and Frobenius-projective structures defined on an algebraic curve in positive characteristic. A Frobenius-projective structure is an analogue of a complex projective structure on a Riemann surface, which was introduced by Y. Hoshi. Such an additional structure has some equivalent objects, e.g., a dormant $\mathrm{PGL}_2$-oper and a projective connection having a full set of solutions. The main result of the present paper provides a canonical construction of a Frobenius-constant quantization on the cotangent space minus the zero section on an algebraic curve by means of a Frobenius-projective structure. It may be thought of as a positive characteristic analogue of a result by D. Ben-Zvi and I. Biswas. Finally, we give a higher-dimensional variant of this result, as proved by I. Biswas in the complex case.