Super-rigidity for Holomorphic Mappings between Hyperquadrics with Positive Signature

In this paper, we study holomorphic mappings from a piece of a real hyperquadric with positive signature into a hyperquadric in a complex space of larger dimension. We will prove that, unlike in the case of Heisenberg hypersurfaces (i.e. hyperquadrics with 0-signature), the maps possess strong super-rigidity properties. This phenomenon is somewhat analogous to that encountered in the study of holomorphic maps between irreducible bounded symmetric domains of rank at least two (see e.g. the book of Mok [Mok] for results and extensive references on this matter.) We first state our results in the context of holomorphic mappings between classical domains in complex projective spaces of different dimensions. For 0 ≤ ` < n, denote by B` the domain in CP n given by B` := {[z0, · · · , zn] ∈ CP : |z0| + · · ·+ |z`| > |z`+1| + · · ·+ |zn|}. For 0 ≤ k ≤ m, let E(k,m) denote the m ×m diagonal matrix with its first k diagonal elements −1 and the rest +1, and define