A boundedness theorem for nearby slopes of holonomic -modules

Abstract

Using twisted nearby cycles, we define a new notion of slopes for complex holonomic${\mathcal{D}}$-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${\mathcal{E}}_{t}$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${\mathcal{E}}_{t}$. This generalizes the regularity of the Gauss–Manin connection proved by Griffiths, Katz and Deligne.