Linearization of holomorphic germs with quasi-Brjuno fixed points

Abstract

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.

20 pages; added some statements and an example to clarify the context