Rigidity of Proper Holomorphic Maps between Type-I Irreducible Bounded Symmetric Domains

Abstract

Abstract We study proper holomorphic maps between type-$\textrm{I}$ irreducible bounded symmetric domains. In particular, we obtain rigidity results for such maps under certain assumptions. More precisely, let $f:D^{\textrm{I}}_{p,q}\to D^{\textrm{I}}_{p^{\prime },q^{\prime }}$ be a proper holomorphic map, where $p\ge q\ge 2$ and $q^{\prime }<\min \{2q-1,p\}$. Then, we show that $p^{\prime }\ge p$ and $q^{\prime }\ge q$. Moreover, we prove that there exist automorphisms $\psi $ and $\Phi $ of $D^{\textrm{I}}_{p,q}$ and $D^{\textrm{I}}_{p^{\prime },q^{\prime }}$, respectively, such that $f=\Phi \circ G_h\circ \psi $ for some map $G_h:D^{\textrm{I}}_{p,q}\to D^{\textrm{I}}_{p^{\prime },q^{\prime }}$ defined by $$G_h(Z):= \left [\begin{array}{cc} Z & \textbf{0}\\ \textbf{0} & h(Z) \end{array}\right ]$$ for all $Z\in D^{\textrm{I}}_{p,q}$, where $h:D^{\textrm{I}}_{p,q}\to D^{\textrm{I}}_{p^{\prime }-p,q^{\prime }-q}$ is a holomorphic map.