Abstract:
In the theory of $\mathcal{D}$-modules over the field $\mathbb C$, the classical theorem of Gabber
states that for a $\mathcal{D}$-module $M$ on a variety $X$, its singular support $SS(M)$ is a
coisotropic subvariety of $T^*_X$; in particular, the singular support of a
holonomic $M$ is Lagrangian. Kontsevich proposed studying $p$-supports defined via
reduction of $\mathcal{D}$-modules to positive characteristic; the usual (singular)
support is then a degeneration of the $p$-support. We will discuss the proof
(following Thomas Bitoun's article "On the $p$-supports of a holonomic $\mathcal{D}$-module") of the Lagrangianity of $p$-supports of holonomic $\mathcal{D}$-modules for all
sufficiently large primes $p$, as well as the deduction of the Lagrangianity of
the singular support from it. The proof is notable for the fact that the main
arithmetic result is obtained using an argument from Hodge theory. I will
begin by recalling some general facts from the theory of $\mathcal{D}$-modules in
characteristics $0$ and $p$.
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