  We study deformations of pairs (X,D), with X smooth projective variety and D
a smooth or a normal crossing divisor, defined over an algebraically closed
field of characteristic 0. Using the differential graded Lie algebras theory
and the Cartan homotopy construction, we are able to prove in a completely
algebraic way the unobstructedness of the deformations of the pair (X,D) in
many cases, e.g., whenever (X,D) is a log Calabi-Yau pair, in the case of a
smooth divisor D in a Calabi Yau variety X and when D is a smooth divisor in
|-m K_X|, for some positive integer m.
