On the $p$-supports of a holonomic $\mathcal {D}$-module (Q1741565)

Let \(Y\) a smooth variety over a perfect filed and let \(D_Y\) be its sheaf of crystalline differential operators. It was shown by \textit{R. Bezrukavnikov} et al., [Ann. Math. (2), 167, 945--991, (2008; Zbl 1220.17009)] that such a sheaf can be seen as an Azumaya algebra over \(T_{Y'}^*\), the tangent sheaf of the Frobenius twist \(Y'\) of \(Y\) (we will write \(F\) to denote the Frobenius operator). This allows us to consider a support of any \(D_Y\)-module \(M\), thinking of it as a sheaf on \(T_{Y'}^*\). This is what the author of the paper under review defines to be the \(p\)-support of \(M\). This interesting and well-written work uses a remarkable amount of concepts and techniques in order to provide important results on the \(p\)-support of a holonomic \(\mathcal{D}\)-module (over a field of characteristic zero) as \(p\) varies and show that \(p\)-support and singular support of a holonomic \(\mathcal{D}\)-module are two closely related notions. Namely, let \(S\) be an integral scheme dominant and of finite type over \(\operatorname{Spec}\mathbb{Z}\), and let \(X\) be a smooth \(S\)-scheme of relative dimension \(n\) (so that the fibers of \(X\) over the closed points of \(S\) play the role of \(Y\) above). Let also \(M\) be a coherent left module over the sheaf of crystalline relative differential operators \(D_{X/S}\) such that the generic fiber \(M_\mu\) of \(M\) is a nonzero holonomic \(\mathcal{D}_{X_\mu}\)-module. The main theorem of the paper asserts that there is an open dense subset \(U\) of \(S\) such that for all closed points \(s\) in \(U\), the \(p\)-support of \(M_s\) is a Lagrangian subvariety of \(T_{X'_s}^*\). As a corollary, the author gives a new proof of the involutivity of the characteristic variety of a holonomic \(\mathcal{D}\)-module by reduction modulo large primes. Another important result is that the \(p\)-supports of the fibers \(M_s\) are generically equidimensional of dimension \(n\), proved by virtue of a clever use of purity of coherent sheaves in Section 3. The proof of the main theorem above is given piecemeal, starting with a reduction to the case \(X/S=\mathbb{A}_S^n\) in Section 2. In such a context, let \(s\) be a point of a dense open subset of \(S\) and let \(z\) be a generic point of an irreducible component of the \(p\)-support of \(M_s\). In Section 4, the author obtains bounds on the degree of \(\overline{\{z\}}\subset\mathbb{P}_{k(s)}^{2n}\) and the so-called rank \(rk_z(M_s):=\dim_{k(z)}((F_*M_s)_z\otimes k(z))\) in terms of the multiplicity of the generic fiber \(M_\mu\). Finally, the author proves the main theorem using all the previous results, as well as the splitting of the Azumaya algebra \(F_*D_{X_s}\) on the regular locus of the \(p\)-support of \(M_s\) and an interesting study of the \(p\)-curvature operator, arising from the Cartier operator on the closed forms of \(F_*(\Omega_{X_s})\).