A Boundedness theorem for nearby slopes of holonomic D-modules

Using twisted nearby cycles, we define a new notion of slopes for complex holonomic D-modules. We prove a boundedness result for these slopes, study their functoriality and use them to characterize regularity. For a family of (possibly irregular) algebraic connections E_t parametrized by a smooth curve, we deduce under natural conditions an explicit bound for the usual slopes of the differential equation satisfied by the family of irregular periods of the E_t. This generalizes the regularity of the Gauss-Manin connection proved by Katz and Deligne. Finally, we address some questions about analogues of the above results for wild ramification in the arithmetic context.