Decomposition factors of D-modules on hyperplane configurations in general position

Abstract:

Let $\alpha _{1},...,\alpha _{m}$ be linear functions on $\mathbb {C}^{n}$ and ${X=\mathbb {C}^{n}\setminus V(\alpha )},$ where $\alpha =\prod _{i=1}^{m}\alpha _{i}$ and ${V(\alpha )=\{p\in \mathbb {C}^{n}:\alpha (p)=0\}}$. The coordinate ring ${\mathcal {O}_{X}}=\mathbb {C}[x]_{\alpha }$ of ${X}$ is a holonomic $A_{n}$-module, where $A_{n}$ is the $n$-th Weyl algebra, and since holonomic $A_{n}$-modules have finite length, ${\mathcal {O}_{X}}$ has finite length. We consider a “twisted” variant of this $A_{n}$-module which is also holonomic. Define ${M_{\alpha }^{\beta }}$ to be the free rank 1 $\mathbb {C}[x]_{\alpha }$-module on the generator $\alpha ^{\beta }$ (thought of as a multivalued function), where $\alpha ^{\beta }=\alpha _{1}^{\beta _{1}}...\alpha _{m}^{\beta _{m}}$ and the multi-index $\beta =(\beta _{1},...,\beta _{m})\in \mathbb {C}^{m}$. It is straightforward to describe the decomposition factors of ${M_{\alpha }^{\beta }}$, when the linear functions $\alpha _{1},...,\alpha _{m}$ define a normal crossing hyperplane configuration, and we use this to give a sufficient criterion on $\beta$ for the irreducibility of ${M_{\alpha }^{\beta }}$, in terms of numerical data for a resolution of the singularities of $V(\alpha ).$