Holomorphic Linearization of Commuting Germs of Holomorphic Maps

Let f(1),..., f(h) be h >= 2 germs of biholomorphisms of C-n fixing the origin. We investigate the shape that a (formal) simultaneous linearization of the given germs can have, and we prove that if f (1),aEuro broken vertical bar,f (h) commute and their linear parts are almost 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 linearizable. This answers a multi-dimensional version of a problem raised by Moser.