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 $T_{Y^{\prime }}^{\ast },$ the cotangent space of the Frobenius twist $Y^{\prime }$ of $Y.$ Thus to a sheaf of modules $M$ over $D_Y$ one can assign a closed subvariety of $T_{Y^{\prime }}^{\ast },$ called the $p$-support, namely the support of $M$ seen as a sheaf on $T_{Y^{\prime }}^{\ast }.$ 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 $p.$

arXiv commentThe article has been rewritten with much improved exposition as well as some additional results, e.g. Corollary 6.3.1. This is the final version, accepted for publication in Inventiones Mathematicae