A Note on Schwarzian Derivatives and Sugiyama-Yasuda Locally Exact Differentials

Abstract

One main purpose of the present paper is to reorganize, in terms of the notion of a \textit{Schwarz system}, a proof, by means of \textit{Schwarzian derivatives}, of the existence of complex projective structures on compact hyperbolic Riemann surfaces and a proof, by means of \textit{Sugiyama-Yasuda locally exact differentials}, of the existence of Frobenius-projective structures of level two on projective smooth curves in characteristic two. Moreover, we also construct quasi-Schwarz systems for certain Frobenius-affine and Frobenius-projective structures on projective smooth curves in positive characteristic.