Skip to main content

Admissibility of Local Systems for some Classes of Line Arrangements

  • Nguyen Tat Thang (a1)

Let 𝒜 be a line arrangement in the complex projective plane 2 and let M be its complement. A rank one local system 𝓛 on M is admissible if, roughly speaking, the cohomology groups H m(M, 𝓛) can be computed directly from the cohomology algebra H*(M, ℂ). 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 𝒱 1(M) and the Resonance variety 𝓡 1(M).

Hide All
[1] Arapura, D., Geometry of cohomology support loci for local systems. I. J. Algebraic Geom. 6 (1997), no. 3, 563597.
[2] Beauville, A., Annulation du H1 pour les fibr´es en droites plats. In: Complex algebraic varieties (Bayreuth, 1990), Lecture Notes in Math., 1507, Springer, Berlin, 1992, pp. 115.
[3] Choudary, A. D. R., Dimca, A., and Papadima, S., Some analogs of Zariski's theorem on nodal line arrangements. Algebr. Geom. Topol. 5 (2005), 691711.
[4] Dimca, A., On admissible rank one local systems. J. Algebra 321 (2009), no. 11, 31453157.
[5] Dimca, A. and Maxim, L., Multivariable Alexander invariants of hypersurface complements. Trans. Amer. Math. Soc. 359 (2007), no. 7, 35053528.
[6] Dimca, A., Papadima, S., and Suciu, A., Topology and geometry of cohomology jump loci. Duke Math. J. 148 (2009), no. 3, 405457.
[7] Dinh, T., Arrangements de droites et systçmes locaux admissibles. Ph.D. Thesis, Universit´e de Nice, 2009.
[8] Dinh, T., Characteristic varieties for a class of line arrangements. Canad. Math. Bull. 54 (2011), no. 1, 56-67.
[9] Eliyahu, M., Garber, D., and Teicher, M., A conjugation-free geometric presentation of fundamental groups of arrangements. Manuscripta Math. 133 (2010), no. 12, 247271.
[10] Esnault, H., Schechtman, V., and Viehweg, E., Cohomology of local systems on the complement of hyperplanes. Invent. Math. 109 (1992), 557561; Erratum, ibid. 112 (1993), 447.
[11] Falk, M., Arrangements and cohomology. Ann. Comb. 1 (1997), no. 2, 135157.
[12] Falk, M. and Yuzvinsky, S., Multinets, resonance varieties, and pencils of plane curves. Compos. Math. 143 (2007), no. 4, 10691088.
[13] Fan, K.-M., Direct product of free groups as the fundamental group of the complement of a union of lines. Michigan Math. J. 44 (1997), no. 2, 283291.
[14] Green, M. and Lazarsfeld, R., Higher obstructions to deforming cohomology groups of line bundles. J. Amer. Math. Soc. 4 (1991), no. 1, 87103.
[15] Jiang, T. and Yau, S. S.-T., Diffeomorphic types of the complements of arrangements of hyperplanes. Compositio Math. 92 (1994), no. 2, 133155.
[16] Libgober, A. and Yuzvinsky, S., Cohomology of the Orlik-Solomon algebras and local systems. Compositio Math. 121 (2000), no. 3, 337361.
[17] Nazir, S. and Raza, Z., Admissible local systems for a class of line arrangements. Proc. Amer. Math. Soc. 137 (2009), no. 4, 13071313.
[18] Schechtman, V., Terao, H., and Varchenko, A., Local systems over complements of hyperplanes and the Kac-Kazhdan condition for singular vectors. J. Pure Appl. Alg. 100 (1995), no. 13, 93102.
[19] Simpson, C., Subspaces of moduli spaces of rank one local systems. Ann. Sci. E´ cole Norm. Sup. 26 (1993), no. 3, 361401.
Recommend this journal

Email your librarian or administrator to recommend adding this journal to your organisation's collection.

Canadian Mathematical Bulletin
  • ISSN: 0008-4395
  • EISSN: 1496-4287
  • URL: /core/journals/canadian-mathematical-bulletin
Please enter your name
Please enter a valid email address
Who would you like to send this to? *



Full text views

Total number of HTML views: 0
Total number of PDF views: 0 *
Loading metrics...

Abstract views

Total abstract views: 0 *
Loading metrics...

* Views captured on Cambridge Core between <date>. This data will be updated every 24 hours.

Usage data cannot currently be displayed