Tokyo Journal of Mathematics

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

Francisco James LEÓN TRUJILLO

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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 $P$.

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Tokyo J. Math., Volume 29, Number 2 (2006), 429-444.

First available in Project Euclid: 1 February 2007

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Zentralblatt MATH identifier

Primary: 32S22: Relations with arrangements of hyperplanes [See also 52C35]
Secondary: 52C35: Arrangements of points, flats, hyperplanes [See also 32S22] 13N10: Rings of differential operators and their modules [See also 16S32, 32C38] 13D40: Hilbert-Samuel and Hilbert-Kunz functions; Poincaré series


LEÓN TRUJILLO, Francisco James. $\mathcal{D}$-Modules and Arrangements of Hyperplanes. Tokyo J. Math. 29 (2006), no. 2, 429--444. doi:10.3836/tjm/1170348177.

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