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

TL;DR

This paper extends Goldman and Millson's deformation results from Kahler manifolds to smooth proper varieties over finite fields, showing the deformation hulls are defined by quadratic or degree four equations using Weil conjectures.

Contribution

It establishes an analogue of Goldman-Millson's theorem in finite characteristic, replacing Hodge theory with Weil conjectures for deformation functors of fundamental group representations.

Findings

Deformation hulls are quadratic for smooth proper varieties.

Deformation hulls are defined by equations of degree at most four for smooth varieties.

Application of Weil conjectures replaces Hodge theory in the finite field setting.

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.