Deformations of representations of fundamental groups of open Kahler manifolds

Given a compact Kahler manifold, we consider the complement U of a divisor with normal crossings. We study the variety of unitary representations of p i(1)(U) with certain restrictions related to the divisor. We show that the possible singularities of this variety as well as of the corresponding modu li space of irreducible representations are quadratic. In the course of our proof we exhibit a differential graded Lie algebra which controls our defo rmation problem.