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Admissibility of Local Systems for some Classes of Line Arrangements
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- Journal:
- Canadian Mathematical Bulletin / Volume 57 / Issue 3 / 01 September 2014
- Published online by Cambridge University Press:
- 20 November 2018, pp. 658-672
- Print publication:
- 01 September 2014
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- Article
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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 
${{\mathcal{R}}_{1}}\left( M \right)$.
