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Lectures on Orlik-Solomon Algebras

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Arrangements, Local Systems and Singularities

Part of the book series: Progress in Mathematics ((PM,volume 283))

Abstract

The first part of this survey is an elementary introduction into Orlik-Solomon algebras. The remaining part is devoted to more recent results, such as the description of the geometric meaning of the Orlik-Solomon complex in terms of the computation of rank 1 local system cohomology via logarithmic connections.

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References

  1. J. Amorós, M. Burger, K. Corlette, D. Kotschick, D. Toledo, Fundamental groups of compact Kähler manifolds, Math. Surveys Monogr., vol. 44, Amer. Math. Soc., Providence, RI, 1996.

    Google Scholar 

  2. K. Aomoto, Un théoréme du type de Matsushima-Murakami concernant l’intégrale des fonctions multiformes, J. Math. Pures Appl. 52 (1973), 1–11.

    Article  MATH  MathSciNet  Google Scholar 

  3. K. Aomoto, Les équations aux différences linéaires des fonctions multiformes, J. Fac. Sci. Univ. Tokyo, sec. IA 22 (1975), 271–297.

    MATH  MathSciNet  Google Scholar 

  4. D. Arapura, Geometry of cohomology support loci for local systems I, Journal of Algebraic Geometry, 6 (1997), 563–597.

    MATH  MathSciNet  Google Scholar 

  5. G. Barthel, F. Hirzebruch, and T. Höfer, Geradenkonfigurationen und Algebraische Fächen, Aspects of Mathematics, D4. Friedr. Vieweg & Sohn, Braunschweig, 1987.

    Google Scholar 

  6. A. Beauville: Annulation du H 1 pour les fibrés en droites plats, in: Complex algebraic varieties (Bayreuth, 1990), 1–15, Lecture Notes in Math., vol. 1507, Springer, Berlin, 1992.

    Chapter  Google Scholar 

  7. E. Brieskorn, Sur les groupes de tresses, Séminaire Bourbaki 1971/72, LNM 317, Springer Verlag, 1973, 21–44.

    Google Scholar 

  8. N. Budur: Unitary local systems, multiplier ideals, and polynomial periodicity of Hodge numbers, Advances in Math. 221 (2009), 217–250.

    Article  MATH  MathSciNet  Google Scholar 

  9. F. Campana: Ensembles de Green-Lazarsfeld et quotients resolubles des groupes de Kähler, J. Alg. Geometry 10(2001), 599–622.

    MATH  MathSciNet  Google Scholar 

  10. J.I. Cogolludo Agustin, D. Matei, Cohomology algebra of plane curves, weak combinatorial type and formality, arxiv:0711.1951.

    Google Scholar 

  11. D. Cohen and P. Orlik, Arrangements and local systems, Math. Res. Lett. 7 (2000), 299–316.

    MATH  MathSciNet  Google Scholar 

  12. D. Cohen and A. Suciu, Characteristic varieties of arrangements, Math. Proc. Cambridge Phil. Soc 127 (1999), 33–53.

    Article  MATH  MathSciNet  Google Scholar 

  13. P. Deligne, Equations différentielles á points singuliers réguliers, Lecture Notes in Math., 163, Springer, Berlin (1970).

    Google Scholar 

  14. P. Deligne, Théorie de Hodge II, Publ. Math. IHES, 40, 5–57 (1972).

    Google Scholar 

  15. P. Deligne, P. Griffiths, J. Morgan, D. Sullivan, Real homotopy theory of Kähler manifolds, Invent. Math. 29 (1975), no. 3, 245–274.

    Article  MATH  MathSciNet  Google Scholar 

  16. A. Dimca, Sheaves in Topology, Universitext, Springer Verlag, 2004.

    Google Scholar 

  17. A. Dimca, Pencils of Plane Curves and Characteristic Varieties, this volume, 59–82.

    Google Scholar 

  18. A. Dimca and G.I. Lehrer, Purity and equivariant weight polynomials, in: Algebraic Groups and Lie Groups, editor G.I. Lehrer, Cambridge University Press, 1997.

    Google Scholar 

  19. A. Dimca, L. Maxim: Multivariable Alexander invariants of hypersurface complements, Trans. Amer. Math. Soc. 359 (2007), no. 7, 3505–3528.

    Article  MATH  MathSciNet  Google Scholar 

  20. A. Dimca, S. Papadima, A. Suciu, Topology and geometry of cohomology jump loci, Duke Math. J. 148 (2009), no. 3, 405–457.

    Article  MATH  MathSciNet  Google Scholar 

  21. A. Dimca, S. Papadima, A. Suciu, Quasi-Kähler Bestvina-Brady groups, J. Algebraic Geom. 17 (2008), no. 1, 185–197.

    MATH  MathSciNet  Google Scholar 

  22. A. Dimca, S. Papadima, A. Suciu, Alexander polynomials: Essential variables and multiplicities, Int. Math. Research Notices vol. 2008 (2008), no. 3, Art. ID rnm119, 36 pp.

    Google Scholar 

  23. H. Esnault, V. Schechtman, and E. Viehweg, Cohomology of local systems of the complement of hyperplanes, Invent. math. 109 (1992), 557–561; Erratum, ibid. Invent. math. 12 (1993) 447.

    Article  MATH  MathSciNet  Google Scholar 

  24. D. Eisenbud, S. Popoescu, and S. Yuzvinsky, Hyperplane arrangement cohomology and monomials in the exterior algebra, Transactions of AMS 355 (2003), 4365–4383.

    Article  MATH  Google Scholar 

  25. M. Falk, Arrangements and cohomology, Ann. Combin. 1 (1997), 135–157.

    Article  MATH  MathSciNet  Google Scholar 

  26. M. Falk and S. Yuzvinsky,Multinets, resonance varieties, and pencils of plane curves, Compositio Math. 143 (2007), no. 4, 1069–1088.

    MATH  MathSciNet  Google Scholar 

  27. J. Folkman, The homology groups of a lattice, J. Math. and Mech. 15 (1966), 631–636.

    MATH  MathSciNet  Google Scholar 

  28. M. Green, R. Lazarsfeld, Higher obstructions to deforming cohomology groups of line bundles, J. Amer. Math. Soc. 4(1991), no. 1, 87–103.

    Article  MATH  MathSciNet  Google Scholar 

  29. P. Griffiths and J. Harris, Principles of Algebraic Geometry, Wiley Interscience, New York, 1978.

    MATH  Google Scholar 

  30. Y. Kawahara. The non-vanishing cohomology of Orlik-Solomon algebras, preprint, 2005.

    Google Scholar 

  31. 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., vol. 36, Kluwer Acad. Publ., Dordrecht, 2001, pp. 215–254.

    Google Scholar 

  32. A. Libgober, First order deformations of local systems with non vanishing cohomology, Topology Appl. 118 (2002), 159–168.

    Article  MATH  MathSciNet  Google Scholar 

  33. A. Libgober and S. Yuzvinsky. Cohomology of the Orlik-Solomon algebras and local systems, Compositio mathematica, 121 (2000), 337–361.

    Article  MATH  MathSciNet  Google Scholar 

  34. M. A. Marco Buzunariz, Resonance varieties, admissible line combinatorics, and combinatorial pencils, arxiv:math.CO/0505435.

    Google Scholar 

  35. J. W. Morgan: The algebraic topology of smooth algebraic varieties, Inst. Hautes Études Sci. Publ. Math. 48 (1978), 137–204.

    Article  MATH  Google Scholar 

  36. P. Orlik and L. Solomon, Combinatorics and topology of complements of hyperplanes, Invent. math. 56 (1980), 167–189.

    Article  MATH  MathSciNet  Google Scholar 

  37. P. Orlik and H. Terao, Arrangements of Hyperplanes, Springer Verlag, Berlin-Heidelberg-New York, 1992.

    MATH  Google Scholar 

  38. J. Pereira and S. Yuzvinsky, Completely reducible hypersurfaces in a pencil, Advances in Math. 219 (2008), 672–688.

    Article  MATH  MathSciNet  Google Scholar 

  39. V. Schechtman and A. Varchenko, Arrangements of hyperplanes and Lie algebra homology, Invent. Math. 106 (1991), 139–194.

    Article  MATH  MathSciNet  Google Scholar 

  40. V. Schechtman, H. Terao, and A. Varchenko, Local systems over complements of hyperplanes and the Kac-Kazhdan condition for singular vectors, J. Pure Appl. Algebra, 100, 93–102 (1995)

    Article  MATH  MathSciNet  Google Scholar 

  41. C. Simpson, Subspaces of moduli spaces of rank 1 local systems, Ann. Sci. École Norm. Sup. 26(1993), no. 3, 361–401.

    MATH  MathSciNet  Google Scholar 

  42. J. Stipins, On finite k-nets in the complex projective plane, Ph. D. thesis, The University of Michigan, 2007.

    Google Scholar 

  43. M. Wachs and J. Walker, On geometric semilattices, Order 2 (1986), 367–385.

    Article  MATH  MathSciNet  Google Scholar 

  44. S. Yuzvinsky, Realization of finite abelian groups by nets in ℙ2, Compos. Math., 140(6) (2004),1614–1624.

    MathSciNet  Google Scholar 

  45. S. Yuzvinsky, Cohomology of the Brieskorn-Orlik-Solomon algebras, Comm. in Algebra 23 (1995), 5339–5354.

    Article  MATH  MathSciNet  Google Scholar 

  46. S. Yuzvinsky, Orlik-Solomon algebras in algebra, topology, and geometry, Russian Math. Surveys 56 (2001), 294–364.

    Article  MathSciNet  Google Scholar 

  47. S. Yuzvinsky, A new bound on the number of special fibers in a pencil of curves, Proc. Amer. Math. Soc. 137 (2009), 1641–1648.

    Article  MATH  MathSciNet  Google Scholar 

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Author information

Authors and Affiliations

  1. Laboratoire J.A. Dieudonné UMR du CNRS 6621, Université de Nice Sophia Antipolis, Parc Valrose, 06108, Nice Cedex 02, France

    Alexandru Dimca

  2. Sergey Yuzvinsky, University of Oregon, Eugene, OR, 97403, USA

    Sergey Yuzvinsky

Authors
  1. Alexandru Dimca
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  2. Sergey Yuzvinsky
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Editor information

Editors and Affiliations

  1. Institute of Mathematics, Case 7012, 2 place Jussieu, 75251, Paris Cedex 05, France

    Fouad El Zein

  2. Center for Advanced Mathematical Sciences (CAMS), American University of Beirut, P.O. Box 11-236, Beirut, Lebanon

    Fouad El Zein

  3. Department of Mathematics, Northeastern University, 360 Huntington Avenue, Boston, MA, 02115, USA

    Alexandru I. Suciu

  4. Mathematics Department, Galatasaray University, Ciragan Cad. No 36 Besiktas, 34357, Istanbul, Turkey

    Meral Tosun  & A. Muhammed Uludağ  & 

  5. Department of Mathematics, University of Oregon, Eugene, OR, 97405, USA

    Sergey Yuzvinsky

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© 2009 Birkhäuser Verlag Basel/Switzerland

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Dimca, A., Yuzvinsky, S. (2009). Lectures on Orlik-Solomon Algebras. In: El Zein, F., Suciu, A.I., Tosun, M., Uludağ, A.M., Yuzvinsky, S. (eds) Arrangements, Local Systems and Singularities. Progress in Mathematics, vol 283. Birkhäuser Basel. https://doi.org/10.1007/978-3-0346-0209-9_4

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