April 19, 2010

Time: 17:30 - 18:30

Room: CRM C1/028
Adolfo Quirós, Universidad Autónoma de Madrid
Simpson’s correspondence and arithmetic differential operators in  positive characteristic

Abstract 

In 1992, Carlos Simpson established an equivalence between the  category of representations of the fundamental group of a compact  Kähler manifold X and the category of Higgs bundles on X, subject to  certain stability conditions. By the classical Riemann-Hilbert  correspondence, a representation of the fundamental group of X can be  viewed as a locally free sheaf E with integrable connection, and  Simpson's main result, the nonabelian Hodge decomposition, is an  isomorphism between the de Rham cohomology for E and the cohomology of  the corresponding Higgs complex that generalizes the usual Hodge  decomposition.

 Several authors have studied analogues of Simpson’s correspondence in  p-adic or finite characteristic settings. Most notably, Ogus and  Vologodsky developed a theory in characteristic p>0 in which p-  curvature plays the role of the Higgs field. It allowed them to proof  an analogue of the Hodge decomposition extending the results of  Deligne and Illusie.

 We present recent work, done jointly with M. Gros and B. Le Stum,  where we use arithmetic differential operators (of level m) to extend  some of the Ogus-Vologodsky results. In particular, we define the  notion of p^m-curvature and, under certain lifting hypothesis, build a  Frobenius map on the ring of differential operators of level m. We  obtain a splitting of a central completion of this ring and then  derive a Simpson correspondence.