Projective and affine structures in positive characteristic I: Chern class formulas and characterizations of projective spaces

Abstract This paper aims to develop a theory of projective and affine structures on higher dimensional varieties in positive characteristic. This theory deals with Frobenius‐projective and Frobenius‐affine structures, which have been previously investigated in the case where the underlying space is a curve. We first provide a description of such structures in terms of Berthelot's higher level differential operators. That description leads us to obtain a positive characteristic version of Gunning's formulas, which give necessary conditions on Chern classes for the existence of Frobenius‐projective and Frobenius‐affine structures, respectively. Finally, we establish some characterizations of projective spaces using Frobenius‐projective structures.