Arthur Ogus

Higgs bundles, p-curvature, and the Cartier operator

Let f : X --> S be a smooth morphism, let (E,\nabla) be a sheaf of O_X-modules with integrable connections on X/S, and let E_{DR} be its associated De Rham complex. When S is the field C of complex numbers and X is an analytic space, E_{DR} is a resolution of a locally constant sheaf of C-vector spaces. This is no longer true in characteristic p, but it turns out that the cohomology sheaves of E_{DR} can be described in terms of a Higgs bundle created from the p-curvature of the connection. This result generalizes the classical Cartier operator and can perhaps be viewed as a partial analog of Simpson's work on nonabelian Hodge theory. As a consequence, we obtain an algebraic proof of a special case of a recent theorem of Barranikov and Kontsevich.