# Publications of the Research Institute for Mathematical Sciences

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**Volume 56, Issue 2, 2020, pp. 401–430**

**DOI: 10.4171/PRIMS/56-2-5**

Published online: 2020-04-02

Frobenius-Projective Structures on Curves in Positive Characteristic

Yuichiro Hoshi^{[1]}(1) Kyoto University, Japan

In the present paper we study *Frobenius-projective structures* on projective smooth curves in positive characteristic. The notion of Frobenius-projective structures may be regarded as an analogue, in positive characteristic, of the notion of *complex projective structures* in the classical theory of Riemann surfaces. By means of the notion of Frobenius-projective structures we obtain a relationship between a certain rational function, i.e., a *pseudo-coordinate*, and a certain collection of data which may be regarded as an analogue, in positive characteristic, of the notion of *indigenous bundles* in the classical theory of Riemann surfaces, i.e., a *Frobenius-indigenous structure*. As an application of this relationship, we also prove the existence of certain *Frobenius-destabilized* locally free coherent sheaves of rank two.

*Keywords: *Pseudo-coordinate, Frobenius-projective structure, Frobenius-indigenous structure, Frobenius-destabilized bundle, $p$-adic Teichmuller theory

Hoshi Yuichiro: Frobenius-Projective Structures on Curves in Positive Characteristic. *Publ. Res. Inst. Math. Sci.* 56 (2020), 401-430. doi: 10.4171/PRIMS/56-2-5