The deformation theory of representations of the fundamental group of a smooth variety

Details The deformation theory of representations of the fundamental group of a smooth variety The deformation theory of representations of the fundamental group of a smooth variety

2004-01-25

arxiv.org/abs/math/0401344v1

Mathematics

Algebraic Geometry

22 pages

Scientific paper

Consider the functor describing deformations of a representation of the fundamental group of a variety X. This paper is chiefly concerned with establishing an analogue in finite characteristic of a result proved by Goldman and Millson for compact Kahler 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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