Simultaneous linearization of holomorphic germs in presence of resonances

Let $ f_{1}, \dots , f_{m}$ be $ m\ge 2$ germs of biholomorphisms of $\mathbb{C}^{n}$, fixing the origin, with $(\mathrm{d}f_{1})_{O}$ diagonalizable and such that $f_{1}$ commutes with $ f_{h}$ for any $h=2,\dots , m$. We prove that, under certain arithmetic conditions on the eigenvalues of $(\mathrm{d}f_{1})_{O}$ and some restrictions on their resonances, $f_{1}, \dots , f_{m}$ are simultaneously holomorphically linearizable if and only if there exists a particular complex manifold invariant under $f_{1}, \dots , f_{m}$.