Geodesic
Admissibility of Local Systems for some Classes of Line Arrangements
Canadian mathematical bulletin, Tome 57 (2014) no. 3, pp. 658-672

Voir la notice de l'article provenant de la source Cambridge University Press

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 ${{\mathcal{R}}_{1}}\left( M \right)$ .
DOI : 10.4153/CMB-2014-030-8
Mots-clés : 14F99, 32S22, 52C35, 05A18, 05C40, 14H50, local system, line arrangement, characteristic variety, multinet, resonance variety 
Thang, Nguyen Tat. Admissibility of Local Systems for some Classes of Line Arrangements. Canadian mathematical bulletin, Tome 57 (2014) no. 3, pp. 658-672. doi: 10.4153/CMB-2014-030-8
@article{10_4153_CMB_2014_030_8,
     author = {Thang, Nguyen Tat},
     title = {Admissibility of {Local} {Systems} for some {Classes} of {Line} {Arrangements}},
     journal = {Canadian mathematical bulletin},
     pages = {658--672},
     year = {2014},
     volume = {57},
     number = {3},
     doi = {10.4153/CMB-2014-030-8},
     url = {http://geodesic.mathdoc.fr/articles/10.4153/CMB-2014-030-8/}
}
TY  - JOUR
AU  - Thang, Nguyen Tat
TI  - Admissibility of Local Systems for some Classes of Line Arrangements
JO  - Canadian mathematical bulletin
PY  - 2014
SP  - 658
EP  - 672
VL  - 57
IS  - 3
UR  - http://geodesic.mathdoc.fr/articles/10.4153/CMB-2014-030-8/
DO  - 10.4153/CMB-2014-030-8
ID  - 10_4153_CMB_2014_030_8
ER  - 
%0 Journal Article
%A Thang, Nguyen Tat
%T Admissibility of Local Systems for some Classes of Line Arrangements
%J Canadian mathematical bulletin
%D 2014
%P 658-672
%V 57
%N 3
%U http://geodesic.mathdoc.fr/articles/10.4153/CMB-2014-030-8/
%R 10.4153/CMB-2014-030-8
%F 10_4153_CMB_2014_030_8

[1] [1] Arapura, D., Geometry of cohomology support loci for local systems. I. J. Algebraic Geom. 6 (1997), no. 3, 563–597. Google Scholar

[2] [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. 1–15. Google Scholar

[3] [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), 691–711. Google Scholar | DOI

[4] [4] Dimca, A., On admissible rank one local systems. J. Algebra 321 (2009), no. 11, 3145–3157. Google Scholar | DOI

[5] [5] Dimca, A. and Maxim, L., Multivariable Alexander invariants of hypersurface complements. Trans. Amer. Math. Soc. 359 (2007), no. 7, 3505–3528. Google Scholar | DOI

[6] [6] Dimca, A., Papadima, S., and Suciu, A., Topology and geometry of cohomology jump loci. Duke Math. J. 148 (2009), no. 3, 405–457. Google Scholar | DOI

[7] [7] Dinh, T., Arrangements de droites et systçmes locaux admissibles. Ph.D. Thesis, Universit´e de Nice, 2009. Google Scholar

[8] [8] Dinh, T., Characteristic varieties for a class of line arrangements. Canad. Math. Bull. 54 (2011), no. 1, 56-67. Google Scholar | DOI

[9] [9] Eliyahu, M., Garber, D., and Teicher, M., A conjugation-free geometric presentation of fundamental groups of arrangements. Manuscripta Math. 133 (2010), no. 1–2, 247–271. Google Scholar | DOI

[10] [10] Esnault, H., Schechtman, V., and Viehweg, E., Cohomology of local systems on the complement of hyperplanes. Invent. Math. 109 (1992), 557–561; Erratum, ibid. 112 (1993), 447. Google Scholar | DOI

[11] [11] Falk, M., Arrangements and cohomology. Ann. Comb. 1 (1997), no. 2, 135–157. Google Scholar | DOI

[12] [12] Falk, M. and Yuzvinsky, S., Multinets, resonance varieties, and pencils of plane curves. Compos. Math. 143 (2007), no. 4, 1069–1088. Google Scholar

[13] [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, 283–291. Google Scholar | DOI

[14] [14] Green, M. and Lazarsfeld, R., Higher obstructions to deforming cohomology groups of line bundles. J. Amer. Math. Soc. 4 (1991), no. 1, 87–103. Google Scholar | DOI

[15] [15] Jiang, T. and Yau, S. S.-T., Diffeomorphic types of the complements of arrangements of hyperplanes. Compositio Math. 92 (1994), no. 2, 133–155. Google Scholar

[16] [16] Libgober, A. and Yuzvinsky, S., Cohomology of the Orlik-Solomon algebras and local systems. Compositio Math. 121 (2000), no. 3, 337–361. Google Scholar | DOI

[17] [17] Nazir, S. and Raza, Z., Admissible local systems for a class of line arrangements. Proc. Amer. Math. Soc. 137 (2009), no. 4, 1307–1313. Google Scholar | DOI

[18] [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. 1–3, 93–102. Google Scholar | DOI

[19] [19] Simpson, C., Subspaces of moduli spaces of rank one local systems. Ann. Sci. E´ cole Norm. Sup. 26 (1993), no. 3, 361–401. Google Scholar

Cité par Sources :