Dedicated to Professor Yum-Tong Siu on the occasion of his 60th birthday 1. Introduction. In this paper, we are concerned with a rigidity problem for holomorphic maps between balls in complex spaces of different dimensions. Write B n and B^ for the unit balls in C n and C^, respectively. Let JP be a proper holomorphic map from B n into B^. We say that F is ^-linear if for any point p G B n , there is an affine complex subspace Sp, that passes through p and is of dimension K, such that for any affine complex line L contained in Sp, F(L H B n ) is contained in an affine complex line in C N . Our main result is the following:
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