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

2004/01/25 by J. P. Pridham, Pridham, J. P.
Mathematics · #14B12 #14F35 #Advanced Algebra and Geometry #Algebraic Geometry (math.AG) #Algebraic Geometry and Number Theory #FOS: Mathematics #Homotopy and Cohomology in Algebraic Topology

paper · pdf · doi:10.48550/arxiv.math/0401344

openalex publication_date 2004/01/25 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

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.

Citations

• Resolution of Singularities of an Algebraic Variety Over a Field of Characteristic Zero: I
• Deformations via Simplicial Deformation Complexes

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