Fedosov quantization in positive characteristic

Fedosov quantization in algebraic context

We consider the problem of quantization of smooth symplectic varieties in the algebro-geometric setting. We show that, under appropriate cohomological assumptions, the Fedosov quantization procedure goes through with minimal changes. The assumptions are satisfied, for example, for affine and for projective varieties. We also give a classification of all possible quantizations.

Quantization of restricted Lagrangian subvarieties in positive characteristic

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.

Quantization on Algebraic Curves with Frobenius-Projective Structure

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…

通过正特征实现Hitchin可积系统的量子化

Quantization of Hitchin integrable system via positive characteristic

针对Beilinson与Drinfeld关于Hitchin可积系统量子化及自守D-模存在性的结论，本文针对G=GL(n)情形，通过约化到正特征、借助泛朗兰兹对偶给出简短证明，并补充了opers空间上p-曲率映射性质的证明。