  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 $f_1,..., f_m$.
