On a semi-rigidity property for holomorphic map

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On the third gap for proper holomorphic maps between balls (2012, background): “(By a result in [19], this codimension is the same as the geometric rank of the map.”
A new gap phenomenon for proper holomorphic mappings from B^n into B^N (2006, methodology): “By the results of [ Hu03 ] and [HJX05], such a map must be rational and (n 0)-linear.”
Super-rigidity for holomorphic mappings between hyperquadrics with positive signature (2005, background): “We refer the reader to the second author’s papers [15], [16] for more references.”