Rigidity and Holomorphic Segre Transversality for Holomorphic Segre Maps

Let Hn and HN denote the complexifications of Heisenberg hypersurfaces in Cn and CN , respectively. We show that non-degenerate holomorphic Segre mappings from Hn into HN with N ≤ 2n− 2 possess a partial rigidity property. As an application, we prove that the holomorphic Segre non-transversality for a holomorphic Segre map from Hn into HN with N ≤ 2n− 2 propagates along Segre varieties. We also give an example showing that this propagation property of holomorphic Segre transversality fails when N > 2n− 2.