Multivariate V-filtrations and the Strong Monodromy Conjecture for hyperplane arrangements

Abstract

In this work, we develop a new theory of multivariate V-filtration on D-modules along a simple normal crossing divisor and relate it with Sabbah's multi-filtration. We establish several new structural results and relate them with the Hodge filtration on free-monodromic local systems from geometric representation theory. As an illustrative application, we give a conceptual and very quick proof of the Strong Monodromy Conjecture and its multivariate generalisation for hyperplane arrangements. Along the way, we confirm both the n/d-conjecture of Budur--Mustaţă--Teitler and its multivariate form due to Budur.