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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

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


Authors: Tilahun Abebaw and Rikard Bøgvad
Journal: Proc. Amer. Math. Soc. 140 (2012), 2699-2711
MSC (2010): Primary 32C38, 52C35; Secondary 14F10, 32S22
Published electronically: November 28, 2011
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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 ).$


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Additional Information

Tilahun Abebaw
Affiliation: Department of Mathematics, Addis Ababa University, Ethiopia – and – Stockholm University, SE-10691 Stockholm, Sweden
Email: tabebaw@math.aau.edu.et

Rikard Bøgvad
Affiliation: Department of Mathematics, Stockholm University, SE-10691 Stockholm, Sweden
Email: rikard@math.su.se

DOI: http://dx.doi.org/10.1090/S0002-9939-2011-11127-4
PII: S 0002-9939(2011)11127-4
Keywords: Hyperplane arrangements, D-module theory
Received by editor(s): July 14, 2010
Received by editor(s) in revised form: March 2, 2011
Published electronically: November 28, 2011
Additional Notes: The first author was supported in part by the International Science Program, Uppsala University
Dedicated: Dedicated to the memory of Demissu Gemeda
Communicated by: Lev Borisov
Article copyright: © Copyright 2011 American Mathematical Society