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
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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Additional Information

J. P. Pridham
Affiliation: Trinity College, Cambridge, CB2 1TQ, United Kingdom
Email: J.P.Pridham@dpmms.cam.ac.uk

DOI: https://doi.org/10.1090/S1056-3911-06-00429-2
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.

American Mathematical Society