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A Simpson correspondence in positive characteristic Michel Gros ,

The first two authors had the opportunity to participate in a working group in Rennes dedicated to the work of Arthur Ogus and Vadim Vologodsky on non abelian Hodge theory, which is now published in [8]. This is an analog in positive characteristic p of Simpson’s correspondence over the complex numbers between local systems and a certain type of holomorphic vector bundles that he called Higgs bundles ([9]). Actually, Pierre Berthelot had previous results related to these questions and he used this opportunity to explain them to us. What we want to do here is to extend these results to differential operators of higher level. In the future, we wish to lift the theory modulo some power of p and compare to Faltings-Simpson p-adic correspondence ([4]).