Simultaneous linearization of holomorphic germs in presence of resonances

Let $f_1, ..., f_m$ be $m\ge 2$ germs of biholomorphisms of $\C^n$, fixing the origin, with $(\d f_1)_O$ diagonalizable and such that $f_1$ commutes with $f_h$ for any $h=2,..., m$. We prove that, under certain arithmetic conditions on the eigenvalues of $(\d f_1)_O$ and some restrictions on their resonances, $f_1, ..., f_m$ are simultaneously holomorphically linearizable if and only if there exists a particular complex manifold invariant under $f_1,..., f_m$.