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Let X be a smooth, projective curve of genus g ≥ 2 over an algebraically closed field of characteristic p > 0. I provide a conjectural formula for the degree of the scheme of dormant Projective General Linear PGL(r)-opers on X where r ≥ 2 (I assume that p is greater than an explicit constant depending on g, r). For r = 2, a dormant PGL(2)-oper is a dormant indigenous bundle on X in the sense of Shinichi Mochuzki (and his work provides a formula only for g = 2, r = 2, p ≥ 5, from a different point of view). In 2014, Yasuhiro Wakabayashi has shown that my conjectural formula holds for r = 2, g ≥ 2, and p > 2g - 2 and more recently he has proved the conjecture in all ranks for generic curves of genus at least two. [ABSTRACT FROM AUTHOR]

Let X be a smooth projective curve of genus g ≥ 2 defined over an algebraically closed field k of characteristic p > 0 . For p > r ( r − 1 ) ( r − 2 ) ( g − 1 ) we construct an atlas for the locus of all Frobenius-destabilized bundles of rank r (i.e. we construct all Frobenius-destabilized bundles of rank r and degree zero up to isomorphism). This is done by exhibiting a surjective morphism from a certain Quot-scheme onto the locus of stable Frobenius-destabilized bundles. Further we show that there is a bijective correspondence between the set of stable vector bundles E over X such that the pull-back F ⁎ ( E ) under the Frobenius morphism of X has maximal Harder–Narasimhan polygon and the set of opers having zero p -curvature. We also show that, after fixing the determinant, these sets are finite, which enables us to derive the dimension of certain Quot-schemes and certain loci of stable Frobenius-destabilized vector bundles over X . The finiteness is proved by studying the properties of the Hitchin–Mochizuki morphism; an alternative approach to finiteness has been realized in [3] . In particular we prove a generalization of a result of Mochizuki to higher ranks. [ABSTRACT FROM AUTHOR]

We show that the locally free sheaf B1⊂F*(Ω1X) of locally exact differentials on a smooth projective curve of genus g⩾2 over an algebraically closed field k of characteristic p is a stable bundle. This answers a question of Raynaud. To cite this article: K. Joshi, C. R. Acad. Sci. Paris, Ser. I 338 (2004). [Copyright &y& Elsevier]