Rationality of proper holomorphic maps between bounded symmetric domains of the first kind

Abstract

Let $D_{p,q}$ and $D_{p',q'}$ be irreducible bounded symmetric domains of the first kind with rank $q$ and $q'$, respectively and let $f:D_{p,q}\to D_{p',q'}$ be a proper holomorphic map that extends $C^2$ up to the boundary. In this paper we show that if $q, q'\geq 2$ and $f$ maps Shilov boundary of $D_{p,q}$ to Shilov boundary of $D_{p',q'}$, then $f$ is of the form $f = \imath\circ F$, where $$ F=F_1\times F_2\colon D_{p,q}\to \Omega_1'\times \Omega_2', $$ $\Omega_1'$ and $\Omega_2'$ are bounded symmetric domains, $F_1 \colon D_{p,q}\to \Omega_1'$ is a proper rational map, $F_2:D_{p,q}\to \Omega_2'$ is not proper and $\imath: \Omega_1' \times \Omega'_2 \hookrightarrow D_{p',q'}$ is a holomorphic totally geodesic isometric embedding of a reducible bounded symmetric domain $\Omega_1' \times \Omega_2'$ into $D_{p',q'}$ with respect to canonical K\"ahler-Einstein metrics. Moreover, if $p>q$, then $f$ is a rational map. As an application we show that a proper holomorphic map $f:D_{p,q}\to D_{p',q'}$ that extends $C^\infty$ up to the boundary is a rational map or a totally geodesic isometric embedding with respect to the Kobayashi metrics, if $3\leq q \leq q'\leq 2q-1.$