Journal of Algebraic Geometry

Journal of Algebraic Geometry

Online ISSN 1534-7486; Print ISSN 1056-3911



Deforming $ l$-adic representations of the fundamental group of a smooth variety

Author: J. P. Pridham
Journal: J. Algebraic Geom. 15 (2006), 415-442
Published electronically: February 28, 2006
MathSciNet review: 2219844
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Abstract: There has long been a philosophy that every deformation problem in characteristic zero should be governed by a differential graded Lie algebra (DGLA). This paper develops the theory of Simplicial Deformation Complexes (SDCs) as an alternative to DGLAs. These work in all characteristics, and for many problems can be constructed canonically.

This theory is applied to study the deformation functor for representations of the étale fundamental group of a variety $ X$. We are chiefly concerned with establishing an algebraic analogue of a result proved by Goldman and Millson for compact Kähler manifolds. By applying the Weil Conjectures instead of Hodge theory, we see that if $ X$ is a smooth proper variety defined over a finite field, and we consider deformations of certain continuous $ l$-adic representations of the algebraic fundamental group, then the hull of the deformation functor will be defined by quadratic equations. Moreover, if $ X$ is merely smooth, then the hull will be defined by equations of degree at most four.

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  • 1. Michel André, Méthode simpliciale en algèbre homologique et algèbre commutative, Lecture Notes in Mathematics, Vol. 32, Springer-Verlag, Berlin-New York, 1967 (French). MR 0214644
  • 2. M. A. Batanin, Coherent categories with respect to monads and coherent prohomotopy theory, Cahiers Topologie Géom. Différentielle Catég. 34 (1993), no. 4, 279–304 (English, with French summary). MR 1253172
  • 3. Pierre Deligne, La conjecture de Weil. II, Inst. Hautes Études Sci. Publ. Math. 52 (1980), 137–252 (French). MR 601520
  • 4. Roger Godement, Topologie algébrique et théorie des faisceaux, Actualit’es Sci. Ind. No. 1252. Publ. Math. Univ. Strasbourg. No. 13, Hermann, Paris, 1958 (French). MR 0102797
  • 5. William M. Goldman and John J. Millson, The deformation theory of representations of fundamental groups of compact Kähler manifolds, Inst. Hautes Études Sci. Publ. Math. 67 (1988), 43–96. MR 972343
  • 6. Luc Illusie, Complexe cotangent et déformations. I, Lecture Notes in Mathematics, Vol. 239, Springer-Verlag, Berlin-New York, 1971 (French). MR 0491680
    Luc Illusie, Complexe cotangent et déformations. II, Lecture Notes in Mathematics, Vol. 283, Springer-Verlag, Berlin-New York, 1972 (French). MR 0491681
  • 7. Laurent Lafforgue, Chtoucas de Drinfeld et correspondance de Langlands, Invent. Math. 147 (2002), no. 1, 1–241 (French, with English and French summaries). MR 1875184,
  • 8. Saunders MacLane, Categories for the working mathematician, Springer-Verlag, New York-Berlin, 1971. Graduate Texts in Mathematics, Vol. 5. MR 0354798
  • 9. Marco Manetti, Deformation theory via differential graded Lie algebras, Algebraic Geometry Seminars, 1998–1999 (Italian) (Pisa), Scuola Norm. Sup., Pisa, 1999, pp. 21–48. MR 1754793
  • 10. James S. Milne, Étale cohomology, Princeton Mathematical Series, vol. 33, Princeton University Press, Princeton, N.J., 1980. MR 559531
  • 11. J. P. Pridham, The structure of the pro-$ l$-unipotent fundamental group of a smooth variety, arXiv math.AG/0401378, 2004.
  • 12. -, Deformations via Simplicial Deformation Complexes, arXiv math.AG/ 0311168, 2005.
  • 13. Daniel Quillen, On the (co-) homology of commutative rings, Applications of Categorical Algebra (Proc. Sympos. Pure Math., Vol. XVII, New York, 1968) Amer. Math. Soc., Providence, R.I., 1970, pp. 65–87. MR 0257068
  • 14. Michael Schlessinger, Functors of Artin rings, Trans. Amer. Math. Soc. 130 (1968), 208–222. MR 0217093,
  • 15. Jean-Pierre Serre, Sur la topologie des variétés algébriques en caractéristique 𝑝, Symposium internacional de topología algebraica International symposium on algebraic topology, Universidad Nacional Autónoma de México and UNESCO, Mexico City, 1958, pp. 24–53 (French). MR 0098097
  • 16. Jean-Pierre Serre, Lie algebras and Lie groups, 2nd ed., Lecture Notes in Mathematics, vol. 1500, Springer-Verlag, Berlin, 1992. 1964 lectures given at Harvard University. MR 1176100
  • 17. Carlos T. Simpson, Higgs bundles and local systems, Inst. Hautes Études Sci. Publ. Math. 75 (1992), 5–95. MR 1179076
  • 18. Charles A. Weibel, An introduction to homological algebra, Cambridge Studies in Advanced Mathematics, vol. 38, Cambridge University Press, Cambridge, 1994. MR 1269324

Additional Information

J. P. Pridham
Affiliation: Trinity College, Cambridge, CB2 1TQ, United Kingdom

Received by editor(s): October 18, 2004
Received by editor(s) in revised form: July 11, 2005
Published electronically: February 28, 2006
Additional Notes: The author was supported during this research by Trinity College, Cambridge and by the Isle of Man Department of Education.

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