On the p-supports of a holonomic mathcal {D}-module

Abstract

For a smooth variety Y over a perfect field of positive characteristic, the sheaf D_Y of crystalline differential operators on Y (also called the sheaf of PD-differential operators) is known to be an Azumaya algebra over T^*_{Y'}, the cotangent space of the Frobenius twist Y' of Y. Thus to a sheaf of modules M over D_Y one can assign a closed subvariety of T^*_{Y'}, called the p-support, namely the support of M seen as a sheaf on T^*_{Y'}. We study here the family of p-supports assigned to the reductions modulo primes p of a holonomic mathcal {D}-module. We prove that the Azumaya algebra of differential operators splits on the regular locus of the p-support and that the p-support is a Lagrangian subvariety of the cotangent space, for p large enough. The latter was conjectured by Kontsevich. Our approach also provides a new proof of the involutivity of the singular support of a holonomic mathcal {D}-module, by reduction modulo p.