Simultaneous linearization of holomorphic germs in presence of resonances

Abstract. Let f 1 , . . . , f m be m ≥ 2 germs of biholomorphisms of C n , fixing the origin, with (df 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 (df 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