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

TL;DR

This paper investigates the $p$-supports of holonomic $ abla$-modules over smooth varieties in positive characteristic, proving they are Lagrangian and that the differential operators split on their regular locus, confirming a conjecture by Kontsevich.

Contribution

It establishes that the $p$-supports are Lagrangian subvarieties and that the Azumaya algebra splits on their regular locus, providing a new proof of involutivity via reduction modulo $p$.

Findings

$p$-supports are Lagrangian for large $p$

Azumaya algebra splits on the regular locus of $p$-supports

New proof of involutivity of singular support

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 $p.$