Linearization of holomorphic germs with quasi-Brjuno fixed points

Let f be a germ of holomorphic diffeomorphism of $${\mathbb {C}^{n}}$$ fixing the origin O, with df_O diagonalizable. We prove that, under certain arithmetic conditions on the eigenvalues of df_O and some restrictions on the resonances, f is locally holomorphically linearizable if and only if there exists a particular f-invariant complex manifold. Most of the classical linearization results can be obtained as corollaries of our result.

Let f be a germ of holomorphic diffeomorphism of C-n fixing the origin O, with d f(O) diagonalizable. We prove that, under certain arithmetic conditions on the eigenvalues of d f(O) and some restrictions on the resonances, f is locally holomorphically linearizable if and only if there exists a particular f-invariant complex manifold. Most of the classical linearization results can be obtained as corollaries of our result.