We prove the existence (and give a characterization) of a germ of complex analytic set left invariant by an abelian group of germs of holomorphic diffeomorphisms at a common fixed point. We also give condition that ensure that such a group can be linearized holomorphically near the fixed point. It rests on a “small divisors condition” of the family of linear parts. The second part of this article is devoted to the study families of totally real intersecting $n$-submanifolds of $(\mathbb C^n,0)$. We give some conditions which allow to straighten holomorphically the family. If it is not possible to do this formally, we construct a germ of complex analytic set at the origin which intersection with the family can be holomorphically straightened. The second part is an application of the first.