On the support of relative $D$-modules

TL;DR

This paper studies the fibers of relative D-modules, proving non-vanishing properties for certain classes and providing new proofs of existing conjectures and theorems, along with a diagonal specialization result for Bernstein-Sato ideals.

Contribution

It establishes non-vanishing results for fibers of relative D-modules, offers new proofs of key conjectures and theorems, and introduces a diagonal specialization result for Bernstein-Sato ideals.

Findings

Existence of a dense subset where fibers are non-zero.

Fibers over all points are non-zero for relatively holonomic D-modules.

New proofs of a conjecture by Budur and a theorem by Maisonobe.

Abstract

In this article we investigate the fibers of relative $D$-modules. In general we prove that there exists an open, Zariski dense subset of the vanishing set of the annihilator over which the fibers of a cyclic relative $D$-module are non-zero. Next we restrict our attention to relatively holonomic $D$-modules. For this class we prove that the fiber over every point in the vanishing set of the annihilator is non-zero. As a consequence we obtain new proofs of a conjecture of Budur which was recently proven by Budur, van der Veer, Wu and Zhou, as well as a new proof of a theorem of Maisonobe. Moreover, we also obtain a diagonal specialization result for Bernstein-Sato ideals.