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

Abstract

Let $D_{p,q}$ and $D_{p^{\prime },q^{\prime }}$ be irreducible bounded symmetric domains of the first kind with rank $q$ and $q^{\prime }$, respectively and let $f:D_{p,q}\to D_{p^{\prime },q^{\prime }}$ be a proper holomorphic map that extends $C^2$ up to the boundary. In this paper we show that if $q,q^{\prime }\ge 2$ and $f$ maps Shilov boundary of $D_{p,q}$ to Shilov boundary of $D_{p^{\prime },q^{\prime }}$, then $f$ is of the form $f=ı\circ F$, where $$F=F_1\times F_2:D_{p,q}\to {\Omega }_1^{\prime }\times {\Omega }_2^{\prime },$$ ${\Omega }_1^{\prime }$ and ${\Omega }_2^{\prime }$ are bounded symmetric domains, $F_1:D_{p,q}\to {\Omega }_1^{\prime }$ is a proper rational map, $F_2:D_{p,q}\to {\Omega }_2^{\prime }$ is not proper and $ı:{\Omega }_1^{\prime }\times {\Omega }_2^{\prime }\hookrightarrow D_{p^{\prime },q^{\prime }}$ is a holomorphic totally geodesic isometric embedding of a reducible bounded symmetric domain ${\Omega }_1^{\prime }\times {\Omega }_2^{\prime }$ into $D_{p^{\prime },q^{\prime }}$ with respect to canonical Kähler-Einstein metrics. Moreover, if $p>q$, then $f$ is a rational map. As an application we show that a proper holomorphic map $f:D_{p,q}\to D_{p^{\prime },q^{\prime }}$ 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 $3\le q\le q^{\prime }\le 2q-1.$