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, this result generalizes to higher-dimensional varieties, as proved by I. Biswas in the complex case.
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Acknowledgements
We would like to thank the referee for reading carefully his manuscript and giving him various helpful comments. Also, we are grateful for the many constructive conversations we had with all algebraic curves with Frobenius-projective structures, who live in the world of mathematics! Our work was partially supported by the Grant-in-Aid for Scientific Research (KAKENHI No. 18K13385, 21K13770).
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Wakabayashi, Y. Quantization on algebraic curves with Frobenius-projective structure. Lett Math Phys 112, 57 (2022). https://doi.org/10.1007/s11005-022-01550-1
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DOI: https://doi.org/10.1007/s11005-022-01550-1