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Admissibility of Local Systems for some Classes of Line Arrangements

Published online by Cambridge University Press:  20 November 2018

Nguyen Tat Thang
Nguyen Tat Thang*
Affiliation:
Institute of Mathematics, Vietnam Academy of Science and Technology, 10307 Hanoi, Vietnam and (temporary) Mathematical Institute, Tohoku University, 980-8578 Sendai, Japan e-mail: ntthang@math.ac.vn
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Abstract

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Let $\mathcal{A}$ be a line arrangement in the complex projective plane ${{\mathbb{P}}^{2}}$ and let $M$ be its complement. A rank one local system $\mathcal{L}$ on $M$ is admissible if, roughly speaking, the cohomology groups ${{H}^{m}}\left( M,\,\mathcal{L} \right)$ can be computed directly from the cohomology algebra ${{H}^{*}}\left( M,\,\mathbb{C} \right)$ . In this work, we give a sufficient condition for the admissibility of all rank one local systems on $M$ . As a result, we obtain some properties of the characteristic variety ${{\mathcal{V}}_{1}}\left( M \right)$ and the Resonance variety ${{\mathcal{R}}_{1}}\left( M \right)$ .

Keywords

14F9932S2252C3505A1805C4014H50local system, line arrangementcharacteristic varietymultinetresonance variety

Information

Type
Research Article
Information
Canadian Mathematical Bulletin , Volume 57 , Issue 3 , 01 September 2014 , pp. 658 - 672
Copyright
Copyright © Canadian Mathematical Society 2014

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