Smooth Families of Tori and Linear K\"ahler Groups

That short note, meant as an addendum to [CCE14], enhances the results contained in loc. cit. In particular it is proven here that a linear K{"a}hler group is already the fundamental group of a smooth complex projective variety. This is achieved by studying the relative deformation of the total space of a smooth family of tori in an equivariant context.