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On representation varieties of Artin groups, projective arrangements and the fundamental groups of smooth complex algebraic varieties

  • Michael Kapovich
  • John J. Millson
Article

Abstract

We prove that for any affine variety S defined overQ there exist Shephard and Artin groups G such that a Zariski open subset U of S is biregular isomorphic to a Zariski open subset of the character variety X(G, PO(3)) = Hom(G, PO(3))//PO(3). The subset U contains all real points of S. As an application we construct new examples of finitely-presented groups which are not fundamental groups of smooth complex algebraic varieties.

Keywords

Fundamental Group Representation Variety Artin Group Abstract Arrangement Zariski Open Subset 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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References

  1. [AK]
    S. Akbulut, H. King,Topology of Real Algebraic Sets, MSRI Publications,25 (1992), Springer-Verlag.Google Scholar
  2. [ABC]
    J. Amoros, M. Burger, K. Corlette, D. Kotschick, D. Toledo,Fundamental groups of Kähler manifolds, AMS Mathematical Surveys and Monographs,44, 1996.Google Scholar
  3. [AN]
    D. Arapura, M. Nori,Solvable fundamental groups of algebraic varieties and Kähler manifolds, Preprint, June 1997.Google Scholar
  4. [A1]
    V. I. Arnold, Normal forms of functions in neighborhoods of degenerate critical points,In: Lecture Notes of London Math. Soc.,53 (1981),Singularity Theory, 91–131.Google Scholar
  5. [A2]
    V. I. Arnold, Critical points of smooth functions and their normal forms, In:Lecture Notes of London Math. Soc.,53 (1981),Singularity Theory, 132–206.Google Scholar
  6. [Ar]
    M. Artin, On the solutions of analytic equations,Invent. Math.,5 (1968), 277–291.zbMATHCrossRefMathSciNetGoogle Scholar
  7. [AM]
    M. Atiyah, I. MacDonald,Introduction to Commutative Algebra, Addison-Wesley, 1969.Google Scholar
  8. [BiM]
    E. Bierstone, P. Millman, Canonical desingularization in characteristic zero by blowing up maximal strata of a local invariant,Inventiones Math.,128 (1997), 207–320.zbMATHCrossRefGoogle Scholar
  9. [Bo]
    N. Bourbaki,Groupes et algèbres de Lie, Chap. 4 à 6, Masson, 1981.Google Scholar
  10. [BuM]
    R. O. Buchweitz, J. J. Millson,CR-geometry and deformations of isolated singularities, Memoirs of AMS,125, N 597 (1997).Google Scholar
  11. [B]
    E. Brieskorn, Die Fundamentalgruppe des Raumes der regulären Orbits einer endlichen komplexen Spiegelungsgruppe,Inventiones Math.,12 (1971), 57–61.zbMATHCrossRefMathSciNetGoogle Scholar
  12. [BS]
    E. Brieskorn, K. Saito, Artin-Gruppen und Coxeter-Gruppen,Inventiones Math.,17 (1972), 245–271.zbMATHCrossRefMathSciNetGoogle Scholar
  13. [C]
    H. S. M. Coxeter, Finite unitary groups generated by reflections,Abhandlungen aus dem Mathematischen Seminar der Universität Hamburg,31 (1967), 125–135.zbMATHMathSciNetGoogle Scholar
  14. [D1]
    P. Deligne, Théorie de Hodge II,Publications Math. IHES,40 (1971), 5–58.zbMATHMathSciNetGoogle Scholar
  15. [D2]
    P. Deligne, Théorie de Hodge III,Publications Math. IHES,44 (1974), 5–77.zbMATHMathSciNetGoogle Scholar
  16. [DGMS]
    P. Deligne, P. Griffith, J. Morgan, D. Sullivan, Real homotopy theory of Kähler manifolds,Inventiones Math.,29 (1975), 245–274.zbMATHCrossRefGoogle Scholar
  17. [DG]
    M. Demazure, P. Gabriel,Groupes algébriques: Vol. I. Géométrie algébrique, généralités, groupes commutatifs, Paris, Masson, 1970.Google Scholar
  18. [Di]
    A. Dimca,Topics on Real and Complex Singularities, Advanced Lectures in Mathematics, Vieweg, 1987.Google Scholar
  19. [EH]
    D. Eisenbud, J. Harris,Schemes: the language of the modern algebraic geometry, Wadsworth & Brooks/Cole Math. Series, 1992.Google Scholar
  20. [EN]
    D. Eisenbud, W. Neumann,Three-dimensional link theory and invariants of plane curve singularities, Ann. of Math. Stud., Princeton Univ. Press,110 (1985).Google Scholar
  21. [GM]
    W. Goldman, J. J. Millson, The deformation theory of representations of fundamental groups of compact Kähler manifolds,Publications Math. IHES,67 (1988), 43–96.zbMATHMathSciNetGoogle Scholar
  22. [GrM]
    P. Griffiths, J. Morgan,Rational homotopy theory and differential forms, Progress in Mathematics, Birkhäuser, 1981.Google Scholar
  23. [Hai]
    R. Hain, in preparation.Google Scholar
  24. [H]
    R. Hartshorne,Foundations of Projective Geometry, Benjamin Inc., NY, 1967.zbMATHGoogle Scholar
  25. [JM]
    D. Johnson, J. J. Millson, Deformation spaces associated to compact hyperbolic manifolds,In: Discrete Groups in Geometry and Analysis, Papers in honor of G. D. Mostow on his 60-th birthday, R. Howe (ed.), Progress in Mathematics,67 (1987), Birkhäuser, 48–106.MathSciNetGoogle Scholar
  26. [KM1]
    M. Kapovich, J. J. Millson, The relative deformation theory of representations and flat connections and deformations of linkages in constant curvature spaces,Compositio Math.,103 (1996), 287–317.zbMATHMathSciNetGoogle Scholar
  27. [KM2]
    M. Kapovich, J. J. Millson, On the deformation theory of representations of fundamental groups of closed hyperbolic 3-manifolds,Topology,35 (1996), 1085–1106.zbMATHCrossRefMathSciNetGoogle Scholar
  28. [KM3]
    M. Kapovich, J. J. Millson, Hodge theory and the art of paper folding,Publications of RIMS, Kyoto,33 (1997), 1–33.zbMATHMathSciNetGoogle Scholar
  29. [KM4]
    M. Kapovich, J. J. Millson, Moduli Spaces of Linkages and Arrangements,In: Advances in Geometry, J.-L. Brylinski (ed.), Progress in Mathematics,172 (1999), Birkhäuser, 237–270.MathSciNetGoogle Scholar
  30. [KM5]
    M. Kapovich, J. J. Millson,Universality theorems for configuration spaces of planar linkages, Preprint, 1998.Google Scholar
  31. [Ke]
    A. B. Kempe, On a general method of describing plane curves of then-th degree by linkwork,Proc. London Math. Soc.,7 (1875), 213–216.CrossRefGoogle Scholar
  32. [Le]
    H. van der Lek, Extended Artin groups,Proc. of Symp. in Pure Math.,40 (1983), Part 2, p. 117–121.Google Scholar
  33. [Lo]
    E. Looijenga, Invariant theory for generalized root systems,Inventiones Math.,61 (1980), 1–32.zbMATHCrossRefMathSciNetGoogle Scholar
  34. [LM]
    A. Lubotzky, A. Magid,Varieties of representations for finitely generated groups, Memoirs of AMS,336 (1985), N 5.Google Scholar
  35. [Mi]
    J. J. Millson, Rational homotopy theory and deformation problems from algebraic geometry,Proc. of ICM 1990,I, p. 549–558.Google Scholar
  36. [Mn]
    N. Mnev, The universality theorems on the classification problem of configuration varieties and convex polytopes varieties,Lecture Notes in Math.,1346 (1988), 527–543.MathSciNetCrossRefGoogle Scholar
  37. [Mo1]
    J. Morgan, Hodge theory for the algebraic topology of smooth algebraic varieties,Proc. Symp. in Pure Math.,32 (1978), 119–127.Google Scholar
  38. [Mo2]
    J. Morgan, The algebraic topology of smooth algebraic varieties,Publications Math. IHES,48 (1978), 137–204.zbMATHGoogle Scholar
  39. [N]
    P. E. Newstead,Introduction to Moduli Problems and Orbit Spaces, Tata Institute Lecture Notes, 1978.Google Scholar
  40. [Sh]
    G. Shephard, Regular complex polytopes,Proc. London Math. Soc.,2 (1952), 82–97.zbMATHCrossRefMathSciNetGoogle Scholar
  41. [Si]
    C. Simpson, Higgs bundles and local systems,Publications Math. IHES,75 (1992), 5–95.zbMATHMathSciNetGoogle Scholar
  42. [St]
    K. G. C. vonStaudt,Beiträge zur Geometre der Lage, Heft 2, 1857.Google Scholar
  43. [Sul]
    D. Sullivan, Infinitesimal computations in topology,Publications Math. IHES,47 (1977), 269–331.zbMATHMathSciNetGoogle Scholar
  44. [Sum]
    H. Sumihiro, Equivariant completion,J. Math. Kyoto Univ.,14 (1974), 1–28;15 (1975), 573–605.zbMATHMathSciNetGoogle Scholar

Copyright information

© Publications mathématiques de l’I.H.É.S 1998

Authors and Affiliations

  • Michael Kapovich
    • 1
  • John J. Millson
    • 2
  1. 1.Department of MathematicsUniversity of UtahSalt Lake CityUSA
  2. 2.Department of MathematicsUniversity of MarylandCollege ParkUSA

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