Deforming 𝑙-adic representations of the fundamental group of a smooth variety

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 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 X is a smooth proper variety defined over a finite field, and we consider deformations of certain continuous l l -adic representations of the algebraic fundamental group, then the hull of the deformation functor will be defined by quadratic equations. Moreover, if X X is merely smooth, then the hull will be defined by equations of degree at most four.