The deformation theory of representations of the fundamental group of a smooth variety
arXiv:math/0401344
Abstract
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.
18 pages; v2 deleted section 2.4 (integral statements) following error in Thm 2.15 discovered by Daniel Litt