  The goal of the present paper is to lay the foundations for 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 only in
the case where the underlying varieties are curves. As the first step in
expanding the theory, we prove various basic properties on Frobenius-projective
and Frobenius-affine structures and study (the positive characteristic version
of) the classification problem, starting with S. Kobayashi and T. Ochiai, of
varieties admitting projective or affine structures. In the first half of the
present paper, we construct bijective correspondences with several types of
generalized indigenous bundles; one of them is defined in terms of Berthelot's
higher-level differential operators. We also prove the positive characteristic
version of Gunning's formulas, which give necessary conditions on Chern classes
for the existence of Frobenius-projective or Frobenius-affine structures
respectively. The second half of the present paper is devoted to studying, from
the viewpoint of Frobenius-projective and Frobenius-affine structures, some
specific classes of varieties, i.e., projective and affine spaces, abelian
varieties, curves, and surfaces. For example, it is shown that the existence of
Frobenius-projective structures of infinite level gives a characterization of
ordinariness for abelian varieties. Also, we prove that any smooth projective
curve of genus $>1$ admits a Frobenius-projective structure of infinite level,
which induces a representation of the stratified fundamental group.
