Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)



Triples of arrangements and local systems

Author: Daniel C. Cohen
Journal: Proc. Amer. Math. Soc. 130 (2002), 3025-3031
MSC (2000): Primary 32S22; Secondary 52C35, 55N25, 14M12
Published electronically: March 15, 2002
MathSciNet review: 1908926
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: For a triple of complex hyperplane arrangements, there is a well-known long exact sequence relating the cohomology of the complements. We observe that this result extends to certain local coefficient systems, and use this extension to study the characteristic varieties of arrangements. We show that the first characteristic variety may contain components that are translated by characters of any order, thereby answering a question of A. Suciu.

References [Enhancements On Off] (What's this?)

  • 1. K. Aomoto, M. Kita, Hypergeometric Functions (in Japanese), Springer-Verlag, 1994.
  • 2. D. Arapura, Geometry of cohomology support loci for local systems I, J. Alg. Geom. 6 (1997), 563-597. MR 98m:32043
  • 3. D. Cohen, P. Orlik, Arrangements and local systems, Math. Res. Lett. 7 (2000), 299-316. MR 2001i:57040
  • 4. D. Cohen, A. Suciu, Characteristic varieties of arrangements, Math. Proc. Cambridge Phil. Soc. 127 (1999), 33-53. MR 2000m:32036
  • 5. M. Falk, Arrangements and cohomology, Ann. Combin. 1 (1997), 135-157. MR 99g:52017
  • 6. M. Falk, Combinatorial and algebraic structures in Orlik-Solomon algebras, European J. Combin. 22 (2001), 687-698. CMP 2001:16
  • 7. I. M. Gelfand, General theory of hypergeometric functions, Soviet Math. Dokl. 33 (1986), 573-577. MR 87h:22012
  • 8. R. Jozsa, J. Rice On the cohomology ring of hyperplane complements, Trans. Amer. Math. Soc. 113 (1991), 973-981. MR 92c:32044
  • 9. A. Libgober, On the homology of finite abelian coverings, Topology Appl. 43 (1992), 157-166. MR 93e:57003
  • 10. A. Libgober, Characteristic varieties of algebraic curves, in: Applications of algebraic geometry to coding theory, physics and computation (Eilat, 2001), NATO Sci. Ser. II Math. Phys. Chem., 36, Kluwer Acad. Publ., Dordrecht, 2001, pp. 215-254. CMP 2002:05
  • 11. A. Libgober, First order deformations for rank one local systems with a non-vanishing cohomology, Topology Appl. 118 (2002), 159-168.
  • 12. A. Libgober, S. Yuzvinsky, Cohomology of the Orlik-Solomon algebras and local systems, Compositio Math. 21 (2000), 337-361. MR 2001j:52032
  • 13. D. Matei, A. Suciu, Cohomology rings and nilpotent quotients of real and complex arrangements, in: Arrangements - Tokyo 1998, Adv. Stud. Pure Math., vol. 27, Math. Soc. Japan, 2000, pp. 185-215. MR 2002b:32045
  • 14. P. Orlik, H. Terao, Arrangements of hyperplanes, Grundlehren Math. Wiss., vol. 300, Springer-Verlag, 1992. MR 94e:52014
  • 15. P. Orlik, H. Terao, Arrangements and Hypergeometric Integrals, MSJ Mem., vol. 9, Math. Soc. Japan, 2001. CMP 2001:08
  • 16. M. Sakuma, Homology of abelian coverings of links and spatial graphs, Canad. J. Math. 47 (1995), 201-224. MR 96d:57008
  • 17. A. Suciu, Translated tori in the characteristic varieties of complex hyperplane arrangements, Topology Appl. 118 (2002), 209-223.
  • 18. A. Varchenko, Multidimensional Hypergeometric Functions and Representation Theory of Lie Algebras and Quantum Groups, Adv. Ser. Math. Phys., vol. 21, World Scientific, 1995. MR 99i:32029
  • 19. S. Yuzvinsky, Orlik-Solomon algebras in algebra and topology, Russian Math. Surveys 56 (2001), 87-166.

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC (2000): 32S22, 52C35, 55N25, 14M12

Retrieve articles in all journals with MSC (2000): 32S22, 52C35, 55N25, 14M12

Additional Information

Daniel C. Cohen
Affiliation: Department of Mathematics, Louisiana State University, Baton Rouge, Louisiana 70803

Keywords: Arrangement, local system, characteristic variety, translated torus
Received by editor(s): April 9, 2001
Received by editor(s) in revised form: May 23, 2001
Published electronically: March 15, 2002
Additional Notes: Partially supported by Louisiana Board of Regents grant LEQSF(1999-2002)-RD-A-01 and by National Security Agency grant MDA904-00-1-0038
Communicated by: Ronald A. Fintushel
Article copyright: © Copyright 2002 American Mathematical Society