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Cosimplicial Objects in Algebraic Geometry

Chapter
Part of the NATO ASI Series book series (ASIC, volume 407)

Abstract

Let X be an arc-connected and locally arc-connected topological space and let I be the unit interval. Applying the connected component functor to each fibre of the fibration of the total space map(I, X) over X × X, P(w) = (w(0), w(1)), we get a local system of sets (Poincaré groupoid) over X × X. This construction does not have a straightforward generalization to algebraic varieties over any field. Using cosimplicial objects, we propose a generalization for smooth, algebraic varieties over an arbitrary field of characteristic zero. This leads to a definition of an algebraic fundamental group of De Rham type. We partly calculate the Betti lattice in the algebraic fundamental group for the projective line minus three points.

Keywords

Vector Bundle Fundamental Group Hopf Algebra Spectral Sequence Characteristic Zero 
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. [BK]
    A.K. Bousfield and D.M. Kan, Homotopy Limits, completions and localizations, L.N. in Math. 304, Berlin-Heidelberg-New York, Springer, 1972.Google Scholar
  2. [BO]
    P. Berthelot and A. Ogus, Notes on crystylline cohomology, Princeton University Press, 1978.Google Scholar
  3. [BS]
    R. Bott and G. Segal, The cohomology of the vector fields on a manifold, Topology Vol. 16, 1977, pp. 285 - 298.MathSciNetzbMATHCrossRefGoogle Scholar
  4. [C]
    K.T. Chen, Iterated integrals, fundamental groups and covering spaces, Trans. of the Amer. Math. Soc. Vol. 206, 83 - 98, 1975.CrossRefGoogle Scholar
  5. [Dl]
    P. Deligne, Equations Différentielles à Points Singuliers Réguliers, Lecture Notes in Math. 163, Springer-Verlag, 1970.Google Scholar
  6. [D2]
    P. Deligne, Le Groupe Fondamental de la Droite Projective Moins Trois Points, in Galois Groups over Q, 1989, Springer-Verlag.Google Scholar
  7. [G]
    A. Grothendieck, Crystals and the De Rham Cohomology of Schemes, in Dix exposés sur la cohomologie des schèmes, Advanced Studies in Pure Mathematics, Vol. 3, North-Holland Publishing Company, 1968.Google Scholar
  8. [H]
    R. Hain, Mixed Hodge Structures on Homotopy Groups, Bull. A.M.S., 15, 1986, pp. 111 - 114.Google Scholar
  9. [HZ]
    R. Hain, S.Zucker, Unipotent variations of mixed Hodge structure, Invent. Math. 88, 1987, pp. 83 - 124.MathSciNetzbMATHGoogle Scholar
  10. [J]
    V. Jannsen, Mixed Motives and Algebraic Ií-theory, Lecture Notes in Math. 1400, Springer-Verlag, 1990.Google Scholar
  11. [KO]
    N.M. Katz, T. Oda, On the differentiation of De Rham cohomology classes with respect to parameters, J. Math. Kyoto Univ. 8-2 (1968), 199 - 213.MathSciNetGoogle Scholar
  12. [Ma]
    B. Malgrange, IV. Regular connections, after Deligne.Google Scholar
  13. [Mo]
    J. Morgan, The algebraic topology of smooth algebraic varieties, Publ. Math. I.H.E.S. 48, 1978, pp. 137 - 204.zbMATHGoogle Scholar
  14. [N]
    V. Navarro Aznar, Sur la théorie de Hodge-Deligne, Invent. Math. 90, 1987, pp. 11 - 76.MathSciNetzbMATHCrossRefGoogle Scholar
  15. [Wl]
    Z. Wojtkowiak, Mixed Hodge structures in the cohomology of cosimplicial spaces and motives associated to homotopy groups, in preparation.Google Scholar
  16. [W2]
    Z. Wojtkowiak, Monodromy of polylogarithms and cosimplicial spaces.Google Scholar

Copyright information

© Springer Science+Business Media Dordrecht 1993

Authors and Affiliations

  1. 1.Sophia Antipolis Laboratoire de Mathématique U.R.A. au C.N.R.S.Université de NiceNice Cedex 2France

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