Rigidity of proper holomorphic maps between type-$\mathrm{I}$ irreducible bounded symmetric domains

Abstract

We study proper holomorphic maps between type-$\mathrm{I}$ irreducible bounded symmetric domains. In particular, we obtain rigidity results for such maps under certain assumptions. More precisely, let $f:D_{p,q}^{\mathrm{I}}\to D_{p^{\prime },q^{\prime }}^{\mathrm{I}}$ 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 $D_{p,q}^{\mathrm{I}}$ and $D_{p^{\prime },q^{\prime }}^{\mathrm{I}}$ respectively, such that $f=\Phi \circ G_h\circ \psi$ for some map $G_h:D_{p,q}^{\mathrm{I}}\to D_{p^{\prime },q^{\prime }}^{\mathrm{I}}$ defined by \[ G_h(Z):= \begin{bmatrix} Z & {\bf 0}\\ {\bf 0} & h(Z) \end{bmatrix}\quad \forall\; Z\in D^{\mathrm{I}}_{p,q},\] where $h:D_{p,q}^{\mathrm{I}}\to D_{p^{\prime }-p,q^{\prime }-q}^{\mathrm{I}}$ is a holomorphic map.

The proof of Proposition 6.6 in version 1 of the article is amended. This proposition becomes Proposition 6.8 in version 2 of the article