Lagrangianity for log extendable overconvergent $F$-isocrystals

Abstract

In the framework of Berthelot's theory of arithmetic $\mathcal{D}$-modules, we prove that Berthelot's characteristic variety associated with a holonomic $\mathcal{D}$-modules endowed with a Frobenius structure has pure dimension. As an application, we get the lagrangianity of the characteristic variety of a log extendable overconvergent $F$-isocrystal.