Abstract
Let α1, α2,…, αm be linear forms defined on ℂn and $\mathcal{X}=\mathbb{C}^{n}\setminus\bigcup_{i=1}^{m}V(\alpha_{i})$ , where V(αi)={p∈ℂn:αi(p)=0}. The coordinate ring $O_{\mathcal{X}}$ of $\mathcal{X}$ is a holonomic An-module, where An is the nth Weyl algebra and since holonomic An-modules have finite length, $O_{\mathcal{X}}$ has finite length. We consider a “twisted” variant of this An-module which is also holonomic. Define M ${}_{α}^{β}$ to be the free rank-1 ℂ[x]α-module on the generator αβ (thought of as a multivalued function), where $\alpha^{\beta}=\alpha_{1}^{\beta_{1}},\ldots,\alpha_{m}^{\beta_{m}}$ and the multi-index β=(β1,…, βm)∈ℂm. Our main result is the computation of the number of decomposition factors of M ${}_{α}^{β}$ and their description when n=2.
Funding Statement
We would like to thank Jan-Erik Björk and Rolf Källström for their interest and crucial contributions. The first author gratefully acknowledges the support by ISP, Uppsala University.
Citation
Tilahun Abebaw. Rikard Bøgvad. "Decomposition of D-modules over a hyperplane arrangement in the plane." Ark. Mat. 48 (2) 211 - 229, October 2010. https://doi.org/10.1007/s11512-009-0103-7
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