  Given a compact Kaehler manifold, we consider the complement U of a divisor
with normal crossings. We study the variety of unitary representations of the
fundamental group of U with certain restrictions related to the divisor. We
show that the possible singularities of this variety as well as of the
corresponding moduli space of irreducible representations are quadratic. In the
course of our proof we exhibit a differential graded Lie algebra of which
reflects our deformation problem.
