$\mathcal{D}$-Modules and Arrangements of Hyperplanes

Abstract

Let $\mathcal{A}$ be a central arrangement of hyperplanes in $\mathbb{C}^n$ defined by the homogeneous polynomial $d_{\mathcal{A}}$. Let $D_n$ be the Weyl algebra of rank $n$ over $\mathbb{C}$ and let $P=\mathbb{C}[x_1,\ldots ,x_n,d_{\mathcal{A}}^{-1}]$ be the algebra of rational functions on the variety $Y_{\mathcal{A}}=\mathbb{C}^n\setminus \bigcup_{H\in \mathcal{A}}H$. Studying the structure of $P$ as a $D_n$-module we obtain a sequence of new $D_n$-modules. These modules allow us to define useful complexes that determine the De Rham cohomology of $Y_{\mathcal{A}}=\mathbb{C}^n\setminus \bigcup_{H\in \mathcal{A}}H$. Finally we compute the Poincaré series of $P$.