Homotopy abelianity of the DG-Lie algebra controlling deformations of pairs (variety with trivial canonical bundle, line bundle)

We investigate the deformations of pairs (X,L), where L is a line bundle on a smooth projective variety X, defined over an algebraically closed field of characteristic 0. In particular, we prove that the DG-Lie algebra controlling the deformations of the pair (X,L) is homotopy abelian whenever X has trivial canonical bundle, and so these deformations are unobstructed.