Holomorphic linearization of commuting germs of holomorphic maps

Abstract. Let f1,..., fh be h ≥ 2 germs of biholomorphisms of Cn fixing the origin. We investigate the shape a (formal) simultaneous linearization of the given germs can have, and we prove that if f1,..., fh commute and their linear parts are al-most simultaneously Jordanizable then they are simultaneously formally linearizable. We next introduce a simultaneous Brjuno-type condition and prove that, in case the linear terms of the germs are diagonalizable, if the germs commute and our Brjuno-type condition holds, then they are holomorphically simultaneously linerizable. This answers to a multi-dimensional version of a problem raised by Moser. 1