Multidegree for bifiltered D-modules

TL;DR

This paper introduces a new multidegree invariant for bifiltered D-modules, connecting algebraic invariants with characteristic cycles and providing explicit formulas for hypergeometric systems in Cohen-Macaulay cases.

Contribution

It extends the concept of multidegree to bifiltered D-modules, linking it with characteristic cycles and hypergeometric systems, and offers explicit formulas in Cohen-Macaulay cases.

Findings

Defined multidegree for bifiltered D-modules

Connected multidegree with L-characteristic cycles

Provided explicit formulas for hypergeometric systems in Cohen-Macaulay cases

Abstract

In commutative algebra, E. Miller and B. Sturmfels defined the notion of multidegree for multigraded modules over a multigraded polynomial ring. We apply this theory to bifiltered modules over the Weyl algebra D. The bifiltration is a combination of the standard filtration by the order of differential operators and of the so-called V-filtration along a coordinate subvariety of the ambient space defined by M. Kashiwara. The multidegree we define provides a new invariant for D-modules. We investigate its relation with the L-characteristic cycles considered by Y. Laurent. We give examples from the theory of A-hypergeometric systems defined by I. M. Gelfand, M. M. Kapranov and A. V. Zelevinsky. We consider the V-filtration along the origin. When the toric projective variety defined from the matrix A is Cohen-Macaulay, we have an explicit formula for the multidegree of the hypergeometric system.