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Admissible local systems for a class of line arrangements


Authors: Shaheen Nazir and Zahid Raza
Journal: Proc. Amer. Math. Soc. 137 (2009), 1307-1313
MSC (2000): Primary 14C21, 14F99, 32S22; Secondary 14E05, 14H50.
DOI: https://doi.org/10.1090/S0002-9939-08-09661-5
Published electronically: November 6, 2008
MathSciNet review: 2465653
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Abstract | References | Similar Articles | Additional Information

Abstract: A rank one local system $ \mathcal{L}$ on a smooth complex algebraic variety $ M$ is admissible if roughly speaking the dimension of the cohomology groups $ H^m(M,\mathcal{L})$ can be computed directly from the cohomology algebra $ H^{\*}(M,\mathbb{C})$.

We say that a line arrangement $ \mathcal{A}$ is of type $ \mathcal{C}_k$ for some $ k\ge 0 $ if $ k$ is the minimal number of lines in $ \mathcal{A}$ containing all the points of multiplicity at least 3. We show that if $ \mathcal{A}$ is a line arrangement in the classes $ \mathcal{C}_k$ for $ k\leq 2$, then any rank one local system $ \mathcal{L}$ on the line arrangement complement $ M$ is admissible. Partial results are obtained for the class $ \mathcal{C}_3$.


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  • 1. A.D.R. Choudary, A. Dimca, S. Papadima: Some analogs of Zariski's Theorem on nodal line arrangements, Algebraic and Geometric Topology 5(2005), 691-711. MR 2153112 (2006f:32038)
  • 2. D.C. Cohen, A.I. Suciu: Characteristic varieties of arrangements, Math. Proc. Cambridge Philos. Soc. 127(1999), 33-54. MR 1692519 (2000m:32036)
  • 3. A. Dimca: Sheaves in Topology, Universitext, Springer-Verlag, Berlin, 2004. MR 2050072 (2005j:55002)
  • 4. A. Dimca: On admissible rank one local systems, J. Algebra (2008), doi:10.1016/j.jalgebra.2008.01.039.
  • 5. A. Dimca, S. Papadima, A. Suciu: Formality, Alexander invariants, and a question of Serre, math.AT/0512480.
  • 6. A. Dimca, L. Maxim: Multivariable Alexander invariants of hypersurface complements, Trans. Amer. Math. Soc. 359(2007), no. 7, 3505-3528. MR 2299465
  • 7. H. Esnault, V. Schechtman, E. Viehweg: Cohomology of local systems on the complement of hyperplanes, Invent. Math. 109(1992), 557-561. Erratum, ibid. 112(1993), 447. MR 1176205 (93g:32051), MR 1213111 (94b:32061)
  • 8. M. Falk: Arrangements and cohomology, Ann. Combin. 1(1997), no. 2, 135-157. MR 1629681 (99g:52017)
  • 9. A. Libgober, S. Yuzvinsky: Cohomology of the Orlik-Solomon algebras and local systems, Compositio Math. 121(2000), 337-361. MR 1761630 (2001j:52032)
  • 10. P. Orlik, H. Terao: Arrangements of hyperplanes, Springer-Verlag, Berlin, 1992. MR 1217488 (94e:52014)
  • 11. V. Schechtman, H. Terao, A. Varchenko: Local systems over complements of hyperplanes and the Kac-Kazhdan condition for singular vectors, J. Pure Appl. Alg. 100(1995), no. 1-3, 93-102. MR 1344845 (96j:32047)
  • 12. A. Suciu: Translated tori in the characteristic varieties of complex hyperplane arrangements. Arrangements in Boston: A Conference on Hyperplane Arrangements (1999). Topology Appl. 118(2002), no. 1-2, 209-223. MR 1877726 (2002j:32027)

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Additional Information

Shaheen Nazir
Affiliation: Abdus Salam School of Mathematical Sciences, Government College University,68-B New Muslim Town, Lahore, Pakistan
Address at time of publication: Abdus Salam School of Mathematical Sciences, Government College University, 35 C-2 Gulberg III, Lahore, Pakistan
Email: shaheen.nazeer@gmail.com

Zahid Raza
Affiliation: Abdus Salam School of Mathematical Sciences, Government College University, 68-B New Muslim Town, Lahore, Pakistan
Email: zahidsms@gmail.com

DOI: https://doi.org/10.1090/S0002-9939-08-09661-5
Keywords: Admissible local system, line arrangement, characteristic variety
Received by editor(s): January 22, 2008
Received by editor(s) in revised form: June 2, 2008
Published electronically: November 6, 2008
Communicated by: Alexander N. Dranishnikov
Article copyright: © Copyright 2008 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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