Simultaneous linearization of holomorphic germs in presence of resonances

Let $f_1,\mathrm{.}\mathrm{.}\mathrm{.},f_m$ be $m\ge 2$ germs of biholomorphisms of ${\text{C}}^n$, fixing the origin, with $(\text{d}{f}_1{)}_O$ diagonalizable and such that $f_1$ commutes with $f_h$ for any $h=2,\mathrm{.}\mathrm{.}\mathrm{.},m$. We prove that, under certain arithmetic conditions on the eigenvalues of $(\text{d}{f}_1{)}_O$ and some restrictions on their resonances, $f_1,\mathrm{.}\mathrm{.}\mathrm{.},f_m$ are simultaneously holomorphically linearizable if and only if there exists a particular complex manifold invariant under $f_1,\mathrm{.}\mathrm{.}\mathrm{.},f_m$.