Deforming $l$-adic representations of the fundamental group of a smooth variety
Author:
J. P. Pridham
Journal:
J. Algebraic Geom. 15 (2006), 415-442
DOI:
https://doi.org/10.1090/S1056-3911-06-00429-2
Published electronically:
February 28, 2006
MathSciNet review:
2219844
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Abstract |
References |
Additional Information
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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Batanin M. A. Batanin, Coherent categories with respect to monads and coherent prohomotopy theory, Cahiers Topologie Géom. Différentielle Catég. 34 (1993), no. 4, 279–304.
Weil2 Pierre Deligne, La conjecture de Weil. II, Inst. Hautes Études Sci. Publ. Math. (1980), no. 52, 137–252.
Go Roger Godement, Topologie algébrique et théorie des faisceaux, Hermann, Paris, 1958.
GM 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. (1988), no. 67, 43–96.
Ill1 Luc Illusie, Complexe cotangent et déformations. I, Springer-Verlag, Berlin, 1971, Lecture Notes in Mathematics, Vol. 239.
La Laurent Lafforgue, Chtoucas de Drinfeld et correspondance de Langlands, Invent. Math. 147 (2002), no. 1, 1–241.
Mac Saunders Mac Lane, Categories for the working mathematician, Springer-Verlag, New York, 1971, Graduate Texts in Mathematics, Vol. 5.
Man Marco Manetti, Deformation theory via differential graded Lie algebras, Algebraic Geometry Seminars, 1998–1999 (Italian) (Pisa), Scuola Norm. Sup., Pisa, 1999, pp. 21–48, arXiv math.AG/0507284.
Mi James S. Milne, Étale cohomology, Princeton University Press, Princeton, N.J., 1980.
pi1 J. P. Pridham, The structure of the pro-$l$-unipotent fundamental group of a smooth variety, arXiv math.AG/0401378, 2004.
sdc ---, Deformations via Simplicial Deformation Complexes, arXiv math.AG/ 0311168, 2005.
Q 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.
Sch Michael Schlessinger, Functors of Artin rings, Trans. Amer. Math. Soc. 130 (1968), 208–222.
torsion Jean-Pierre Serre, Sur la topologie des variĂ©tĂ©s algĂ©briques en caractĂ©ristique $p$, Symposium internacional de topologĂa algebraica International symposium on algebraic topology, Universidad Nacional AutĂłnoma de MĂ©xico and UNESCO, Mexico City, 1958, pp. 24–53.
Se ---, Lie algebras and Lie groups, second ed., Springer-Verlag, Berlin, 1992, 1964 lectures given at Harvard University.
Simpson Carlos T. Simpson, Higgs bundles and local systems, Inst. Hautes Études Sci. Publ. Math. (1992), no. 75, 5–95.
W Charles A. Weibel, An introduction to homological algebra, Cambridge University Press, Cambridge, 1994.
Additional Information
J. P. Pridham
Affiliation:
Trinity College, Cambridge, CB2 1TQ, United Kingdom
Email:
J.P.Pridham@dpmms.cam.ac.uk
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.
