Linearization of holomorphic germs with quasi-Brjuno fixed points

Let f be a germ of holomorphic diffeomorphism of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathbb {C}^{n}}$$\end{document} fixing the origin O, with d fO diagonalizable. We prove that, under certain arithmetic conditions on the eigenvalues of d fO 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.