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  $f_{1}, \dots , f_{m}$.